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This article is cited in 2 scientific papers (total in 2 papers) Billiards and integrable systems A. T. Fomenkoab, V. V. Vedyushkinaa a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University b Moscow Center for Fundamental and Applied Mathematics
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 5(473), DOI: https://doi.org/10.4213/rm10100 
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    IntroductionThe topological approach to the investigation of integrable finite-dimensional system was largely prompted by Smale’s paper [1], which concerned in particular, the study of the topological properties of common level sets of the energy and area integral for Hamiltonian system. Here is an overview of the sections of our survey, which is devoted to illustrating the use of this method for the analysis of fairly transparent integrable system, namely, billiards. Section 1. The theory of topological classification of integrable sysems developed by Fomenko with his coauthors and students [2]–[6] allows one to classify such systems with two degrees of freedom up to leafwise equivalences of various types of their Liouville foliations, and to list algorithmically all non-degenerate singularities of rank $1$ (Fomenko $3$-atoms) and rank $0$ (foliations in an invariant neighbourhood of a leaf containing non-degenerate equilibria) that can occur in such systems. This is based on an analogue of Morse’s theory (the Morse–Bott theory) developed previously by Fomenko [7], [8] for integrable systems with finitely many degrees of freedom. The main invariants of an integrable Hamiltonian system on a non-singular ($d H \ne 0$) level of energy $Q^3_h\colon H=h$ are two framed graphs, the Fomenko invariant (also called molecule), that is, the base of the Liouville foliation and a local lift of it, and the Fomenko–Zieschang invariant (or marked molecule). Vertices of these graphs are labelled by the types of non-degenerate singularities ($3$-atoms), edges correspond to families of nonsingular Liouville tori, and numerical marks code the result of gluing the foliated level surface $Q^3$ from a system of $3$-atoms, that is, they contain requisite information on the system of gluing diffeomorphisms between boundary tori of $3$-atoms in accordance with the incidence matrix of the Fomenko graph. The above topological invariants can be calculated for many integrable systems in geometry, mechanics, and mathematical physics [6]. If the invariants of two systems coincide in non-singular ranges of energy of these systems, then there is a homeomorphism between the systems which takes almost all closures of solutions of one system to the closures of solutions of the other. Although the systems can be defined by different equations, their topological equivalence means nonetheless that many qualitative properties of motion in them are the same. Note that the same can be said about non-integrable perturbations close to these systems, in view of the KAM-theory and results due to Nekhoroshev (see [9] and [10]). Note that in recent years Fomenko’s theory of topological classification was developed in various directions: trajectory equivalence of integrable systems [11], topological and symplectic classification and coding for non-degenerate singularities of integrable systems [12]–[14], classification of degenerate singularities of rank $1$ and questions relating to their structure stability under perturbations in the class of integrable system [15], systems with noncompact foliations and incomplete flows: see, for instance, [16]. Section 2. One direction in the theory of integrable systems that flourished in recent years was the theory of integrable billiard systems. Under consideration is the system of motion of a ball in some domain (the table) involving reflections from the boundary of the domain. The now classical results due to Bolotin [17], [18], along with recent result of Bialy and Mironov [19], [20], Glutsyuk [21], [22], Sorrentino, Kaloshin, and other authors [23], [24] and related to establishing various versions of Birkhoff’s billiard conjecture [25], show that the fact that the boundary arcs of the table belong to a family of confocal quadrics (or degenerate cases of such quadrics) is not just sufficient [26] but also necessary for the integrability of the system in one sense or another (for example, for polynomial integrability). Although billiard systems are, generally speaking, just piecewise smooth, for many of them we can calculate a piecewise smooth analogue of the Fomenko–Zieschang invariant: regular levels for them are tori and neighbourhoods of singular levels are leafwise homeomorphic to $3$-atoms; moreover, numerical marks and admissible bases on gluing tori which specify these marks are well defined. For planar confocal billiards the invariants were calculated by Dragović and Radnović [27] and, independently, Vedushkina, both in the case of ellipses and hyperbolae [28], [29] and in the case of parabolas [30]. Here and throughout we speak about the Fomenko–Zieschang invariant of a billiard system (omitting the word ‘analogue’) when this invariant is well defined. For planar confocal table there can be only a finite number of different invariants. In other words, using such billiards we can only model integrable system in a finite set of classes of Liouville equivalence. One would believe that for integrable billiard systems (without potential) on planar tables the general topological problem is also ‘finite’. Vedyushkina succeeded in enlarging the class of billiards under investigation in a crucial was by allowing piecewise planar two-dimensional tables glued of planar table along common boundary arcs. The classes of topological (generalized) billiards [31] and billiard books [32] were introduced. Topological billiards are homeomorphic to orientable manifolds (although they are only piecewise planar). Each edge ($1$-cell) is a boundary arc, or precisely two planar tables are glued along it. Billiard books are CW-complexes with $1$-cells (‘spines’) framed by cyclic permutations on the set of incident $2$-cells (‘leaves’ of the book). These permutations define the transition of the billiard ball from one leaf to another after bouncing off the spine in question. Corresponding to each vertex (zero-dimensional cell) there is a commutation condition. In contrast to a topological billiard, a book contains at least one edge (a spine) which is common to at least three $2$-cells (‘sheets’ of the book). The class of topological billiards was fully classified by Vedyushkina [31], [33], both structurally (as CW-complexes) and topologically (that is, the Fomenko–Zieschang invariants were calculated). A structure equivalence between tables was defined, which preserves the topology of the Liouville foliation and admits a continuous deformation of boundary arcs in the class of confocal quadrics. The class of billiard books is much wider; it was shown in [34] that using the operation of ‘bending’ a book the whole class can be reduced to nine sets of books, each of which can be listed algorithmically in accordance with the number of sheets. In [35], on the basis of the results known at that time, Fomenko stated a general conjecture (discussed in greater detail in § 2.3), which makes a suggestion that from the standpoint of the topology of Liouville foliations the class of integrable billiards is ‘not narrower’ that the whole class of integrable systems, namely, an arbitrary class of Liouville equivalence (a Fomenki–Zieschang invariant, that is, a graph with atoms as vertices and numerical marks) can be realized by an appropriate billiard system. Any answer is of interest here: if the conjecture fails, then finding the nature of the obstruction to such a realization is of importance. In the same paper Fomenko stated a few other problems concerning the topology and dynamics of integrable billiard systems. Several theses in Fomenko’s conjecture have already been proved: it was shown that one can realize all non-degenerate $3$-atoms [32], [36], all values of numerical marks [37]–[39], as well as an arbitrary Fomenko invariant without marks [40]. In other words, no ‘component’ of the invariant can alone be an obstruction to realization, and moreover, billiards model all classes of relatively weaker (namely, rough Liouville) equivalence. Note that, apart from modelling all Bott (non-degenerate) saddle $3$-atoms by the class of billiard books due to Vedyushkina and Kharcheva, Kusnetsova has recently been successful in producing examples of non-Bott multi-saddle $3$-atoms (which are direct or almost direct products of a multi-saddle and a circle). In § 2.3 we also state an extended version of part A of Fomenko’s conjecture, Conjecture $\widetilde{{\rm A}}$ which states that arbitrary $3$-atoms whose singular circles can be multisaddle, can be realized. The correspondence between non-Morse multisaddle $2$-atoms with one singular point and chord diagrams was considered by Nikonov [41]. A separate topic is the topology of isoenergy sets $Q^3$ for billiard systems. Kharcheva [42] showed that for any book $Q^3$ is homeomorphic to a smooth 3-manifold. Vedyushkina [43] constructed books with $Q^3$ belonging to the complement of the class of Seifert manifolds to the class of Waldhausen graph manifolds [44], [45], so that for billiard systems the class of isoenergy surfaces $Q^3$ is not confined to Seifert manifolds. A number of recent results was also discussed in [46]–[48]. Although the answer in part C, the most general part of the conjecture (on realizing arbitrary Fomenko–Zieschang invariants) is not yet clear, the following systems in mechanics and mathematical physics have already been modelled by billiards [49]: the Euler and Lagrange tops in every non-singular energy range, and the following systems in certain suitable energy ranges: the Kovalevskaya top, the Joukowsky system (the Euler top with a gyrostat, also called the Joukowsky–Volterra system), and the Chaplygin [50], Clebsch, Steklov, and Sokolov systems (see details in § 2.4). It is interesting that, as these authors discovered in [51], some smooth systems with first integral of high (3 or 4) degree (the Goryachev–Chaplygin and Kovalevskaya systems and the geodesic flows corresponding to them in accordance with the Maupertuis principle, for which the degree of the first integral cannot be lower) can be modelled by, generally speaking, piecewise planar billiard systems with the same first intgral of degree $2$. Here, with each non-singular energy range of the modelled system we must associate its ‘own’ billiard book, which realizes the Liouville foliation of this system in the energy range specified. In this context the paper [53] is also of interest, where billiards in a disc with metric of revolution and a potential, which have an irreducible integral of degree $3$ or $4$, are presented. An important series of results is connected with modelling geodesic flows on 2-surfaces which are Liouville-integrable with first integrals of degree $1$ or $2$. By Kozlov’s famous theorem [54], the Euler number of such a surface must be non- negative, that is, it must be homeomorphic to the sphere $S^2$, the torus $T^2$, the projective plane $\mathbb{R}P^2$, or the Klein bottle $\operatorname{KL}^2$. The canonical form of metrics giving rise to such geodesic flows was known before, and the Fomenko–Zieschang invariants and orbital invariants were calculated for all metrics (the history of the question and the results were thoroughly presented in [6]). Integrable flows on a sphere and a torus were modelled by Vedyushkina and Fomenko [55], [46] by means of integrable circular topological billiards (for a linear first integral) and confocal topological billiards and billiard books (for a quadratic first integral). With each such flow defined by a non-trivial Riemannian metric they associated a piecewise planar table with plane metric inside $2$-cells, and with these planar parts glued isometrically along their boundaries. Section 3. The problem of modelling a system by a single billiard on the whole of the phase space $M^4$ simultaneously (perhaps, away from some exceptional energy levels), is quite natural. The following two questions are straightforward complex versions of questions and results discussed in § 2:
In § 3 we describe in detail the class of evolutionary billiards introduced recently [57], [58] and present result on modelling, by means of such systems, the topology of the (whole) Euler and Largange tops (so that all non-singular foliations on the symplectic phase space $M^4$ of the system are realized by a single evolutionary billiard), the Joukowsky system, and the Kovalevskaya top [59]. In doing this we reveal a not-so-obvious link between the classical Euler and Lagrange system. By letting the foci of the quadrics tend to the centre we obtain a deformation of a family of confocal quadrics into a family of concentric circles and their radii. If we make this transformation with the evolutionary billiard modelling the Euler top, then the set of confocal billiards realizing the Liouville foliations for the Euler top in non-singular energy ranges is transformed into a system of circular billiards realizing the Liouville foliations for the Lagrange top in non-singular energy ranges. In this sense we can say that these two systems are ‘billiard equivalent’. In § 3 we also present a new result: as the foci of the quadrics tend to their centre, the confocal topological billiard realizing the geodesic flow of a globally Liouvillean metric on a torus (which has a quadratic first integral) turns to the circular topological billiard realising a geodesic flow with linear first integral on a torus. In other words, linearly integrable geodesic flows on a torus are billiard equivalent to quadratically integrable geodesic flows on a torus with globally Liouvillean metric. Section 4. In that section we present several short stories devoted to different integrable generalizations of classical billiard systems (including integrable billiards with potential [60], [61], circular billiards in constant magnetic field [62], and billiards with slipping introduced by Fomenko [63]) and list a number of other problems: billiards in the plane with Minkowski metric, adding a potential to such systems, the topology of leaves of Liouville foliation for pseudointegrable confocal billiards with angles of $3\pi/2$, ordered billiard games. For the well-known Kozlov equation [60] describing a condition of integrability for the potential of a billiard inside an ellipse (some solutions of this equation were found by Dragović), Pustovoitov indicated the general form of its solutions (provided that the potential is a polynomial). For billiards with such potentials the problem of the description of the topology of the Liouville foliation on a non-singular energy level is algorithmic. This is a development of Pustovoitov’s previous result [64] on the topology of billiards with potentials of degree $4$. The problem of modelling non-degenerate semilocal singularities of rank $0$, mentioned above, that is, of modelling Liouville foliations in a small four-dimensional neighbourhood of a level with non-degenerate equilibria, was successfully solved using an approach proposed by Kibkalo [65], [66]. To do this one uses billiard books on sheets of which motion proceed in the field of a repulsive Hooke potantial, the same for all sheets. In the most difficult case of saddle-saddle singularities the structure of the required book (the number of sheets and the permutations on gluing edges) can be recovered from the $f_n$-graph of the singularity introduced by Oshemkov [13]. A singularity with $n$ equilibria of focus-focus type on the singular level can be modelled by a billiard book with a repulsive Hooke potential on $n$ copies of a billiard inside a circle glued along their common boundary circle, with the permutation $(1,\dots,n)$: see Vedyushkina, Kibkalo, and Pustovoitov [67]. Pustovoitov also investigated the topology of planar and topological billiards on circular tables in which the motion is performed in a constant magnetic field with induction vector orthogonal to the surface of the table. The bifurcation diagrams of such billiard systems were algorithmically constructed and their Fomenko–Zieschang invariants were calculated. The case of billiards on tables of dimension greater than $2$ was investigated by Belozerov. He constructed the bifurcation diagrams and classified the table domains bounded by pieces of confocal quadrics in $\mathbb{R}^3$ up to weak equivalence [68]. The topological properties of billiards in such domains in the case when a Hooke potential is added are currently under investigation. The question of the integrability of the motion of a ball along an intersection of several confocal quadrics in $\mathbb{R}^3$ was posed by Kibkalo, who answered it in the case of $n-2$ quadrics in $\mathbb{R}^n$. The formulae for the additional integral were found and the class of 2-dimensional ‘elementary’ domains with non-plane metric, from which one can also glue topological billiards and billiard books, was extended. In the general case of intersection of $k$ confocal quadrics in $\mathbb{R}^n$ integrability was shown by Belozerov (this integrability is preserved by an addition of a central Hooke potential). In fact, his result generalizes the classical Jacobi–Chasles theorem (corresponding to $k=1$): any tangent to a phase trajectory of the geodesic flow is also tangent to $n-k-1$ confocal quadrics, which are the same for all points on this trajectory. 1. Integrable systems with two degrees of freedom: invariants of Liouville equivalence and classification of systems. Requisite concepts, theorems, and notationIn this paper we consider smooth non-degenerate integrable Hamiltonian systems which mostly have two or three degrees of freedom, and also piecewise smooth integrable billiards, mostly two- and three-dimensional ones. 1.1. Integrable systems with two degrees of freedom. Basic definitions and theoremsLet $M^{2n}$ be a symplectic manifold with symplectic $2$-form $\omega$, and let $v=\operatorname{sgrad} H$ denote a Hamiltonian vector field with smooth Hamiltonian $H$. The system $v$ is said to be completely integrable in the sense of Liouville if it has a system of functionally independent smooth first integrals $f_1,\dots,f_n$ commuting with respect to the Poisson bracket $\{*,*\}$ on $M$, and all vector fields $\operatorname{sgrad}f_i$ are complete. Definition 1. The partition of $M$ into connected components of joint level sets $T_\xi$ of the integrals $f_1,\dots,f_n$ that arises here is called the Liouville foliation. It consists of regular $n$-dimensional leaves filling almost the whole of $M^{2n}$, and of singular leaves filling a subset of measure zero. (We repeat that all leaves are connected.) The following fundamental result (sometimes called Liouville’s theorem; see [69] by Dubrovin, Krichever, and Novikov) describes fully the topology, dynamics, and symplectic geometry of a completely integrable system in a neighbourhood of a regular leaf. The reader can find various proofs of this result, for instance, in the monographs by Dubrovin, Novikov, and Fomenko [70] and Bolsinov and Fomenko [6]. Theorem 1. Consider a Liouville-integrable system, and let $T_\xi$ be a regular $n$- dimensional level set. Then this set is a smooth Lagrangian submanifold which is invariant under the action of the fields (flows) $\operatorname{sgrad}H,\operatorname{sgrad}f_1,\dots$, and $\operatorname{sgrad}f_n$. 1) If $T_\xi$ is compact and connected, then it is diffeomorphic to an $n$-torus, which is called a Liouville torus. 2) In a neighbourhood $U^{2n}$ of the torus $T_\xi$ the Liouville foliation is trivial, that is, it is diffeomorphic to the Cartesian product of $T^n$ and the disc $D^n$. 3) In the neighbourhood indicated in part 2) there exist coordinates $s_1,\dots,s_n$, $\phi_1,\dots,\phi_n$, called ‘action-angle coordinates’ which are functions of the original coordinates of the system. Here the $s_i$ are coordinates on $D^n$, and the $\phi_i$ are the standard angle coordinates on the torus $T_\xi$. 4) In these coordinates the symplectic structure is constant and canonical and the Hamiltonian flow $v$ (as well as the flows $\operatorname{sgrad}f_i$) straightens up on each Liouville torus in $U $, so that $\dot{s}_i=0$, $\dot{\phi}_i=q_i(s_1,\dots,s_n)$, $i=1,\dots,n$. This means that on each torus the flow $v$ defines a conditionally periodic motion, its trajectories and straight-line windings, which can be rational or irrational. (The corresponding Liouville tori are also called resonance or non-resonance.) In the analytic case almost all Liouville tori are non-resonance, that is, the windings on them are irrational. Hence such a torus is the closure of an integral trajectory lying on it. In this sense almost all Liouville tori present the closures of solutions of the system $v$. The system is said to be non-resonance if almost all of its Liouville tori are non-resonance. Definition 2. For systems with two degrees of freedom on $M^4$ we let $Q_h$ denote an isoenergy surface, that is, the level surface $H=h=\operatorname{const}$. Almost all isoenergy surfaces are regular, because $dH$ is distinct from zero on them, so that such $Q^3_h$ are smooth three-dimensional submanifolds of $M$. In what follows we assume that all the $Q_h$ are compact. Let $f$ denote an additional integral of the system which is independent of $H$, and let $F\colon M^4 \to\mathbb{R}^2$ be the moment map $F(x)=(H(x),f(x))$. Let $\sigma$ be the bifurcation diagram, that is, the image of the set of critical points of the moment map in the plane $\mathbb{R}^2$; typically, $\sigma$ consists of piecewise smooth curves and isolated points. Restricting $f$ to a regular surface $Q^3_h$ we obtain a smooth function and a map $f\colon Q\to \mathbb{R}^1$. The integral $f$ is said to be Bott or non-degenerate on $Q$ if all of its critical submanifolds (submanifolds of critical points of $f$) are non-degenerate. This means that the restriction of $f$ to a transversal platform to a critical submanifold is a Morse function. We say that the system $v$ is non-degenerate on $Q$ if all critical submanifolds are non-degenerate. They can only be one- or two-dimensional. In what follows we assume that the systems under consideration are non-degenerate and have only critical circles on regular surfaces $Q^3_h$. Definition 3. Two integrable systems $v_1$ and $v_2$ on $M^4_1$ and $M^4_2$ (on $Q^3_1$ and $Q^3_2$, respectively) are said to be Liouville equivalent if there exists a diffeomorphism of $M^4_1$ onto $M^4_2$ (of $Q^3_1$ onto $Q^3_2$ respectively), that takes the Liouville foliation of the first system to the Liouville foliation of the second and preserves the orientations of the $3$-manifolds and all critical circles (induced by the flows $v_1$ and $v_2$). Recall that critical circles of first integrals are closed trajectories of the system, so they have natural orientations. Since almost all Liouville tori are typically (in the generic case) the closures of integral trajectories, we can say that for Liouville equivalent systems almost all integral trajectories have ‘the same’ closures. Definition 4. The base of a Liouville foliation (or a bifurcation complex) is the topological space of its (connected) leaves with the usual quotient topology, that is, the topological space whose points are leaves of the Liouville foliation (each leaf, whether regular or singular, is replaced by a point). In the general case bifurcation complexes and their properties (including the description of the bifurcations of the foliation on ‘faces’ of the complex) were introduced by Fomenko [2], [71] in 1988. Typically, a bifurcation complex is not just a Hausdorff space, but even a cell complex. For systems with two degrees of freedom on $M^4$ it has dimension two. The bifurcation complex (base) of the Liouville foliation on an isoenergy $3$-surface $Q^3_h$ is a one-dimensional graph. Importantly, in a non-resonance (generic) case the bifurcation complex does not depend on the choice of the integral of the integrable system; it is defined by the vector field $\operatorname{sgrad}H$ alone; see [2] and [71]. Recall that almost all Liouville tori are the closured of irrational integral trajectories. Definition 5. Two integrable systems are said to be roughly Liouville equivalent if there exists a homeomorphism between the bases (bifurcation complexes) of the corresponding Liouville foliations that lifts locally (in a neighbourhood of each point in the base) to a leafwise homeomorphism of the Liouville foliations. 1.2. Critical points and non-degenerate singularities. $2$-atoms $f$-graphsLet $f$ be a Morse function on a closed two-dimensional surface $X^2$. Definition 6. A Morse $2$-atom is a neighbourhood $P^2$ of a critical level $f(x)=c$ given by an inequality $c-\varepsilon \leqslant f \leqslant c+\varepsilon$ for a sufficiently small $\varepsilon$, foliated by level sets of $f$ and considered up to leafwise equivalence. If the critical value $c$ is a local maximum or a minimum, then this atom is called an atom $A$. If $c$ is a saddle value, then the $2$-atom is said to be saddle. An atom is said to be simple if the Morse function is simple, that is, there is a unique critical point on the critical level. Other atoms are said to be complex. An atom is said to be orientable or non-orientable depending in whether the surface $P^2$ is orientable or not. The genus of a $2$-atom is by definition the genus of the closed $2$-surface obtained from $P^2$ by attaching discs to all of its boundary circles. An algorithm for constructing a complete list of Morse $2$-atoms was described in [6]. To each fixed value of the complexity of a $2$-atom, that is, to the number of singular points on it there corresponds a finite list of Morse $2$-atoms with this complexity. There are precisely three atoms with complexity 1 (the minimax atom $A$, the orientable saddle atom $B$ and the non-orientable saddle atom $\widetilde{B}$). All atoms with complexity $2$ or higher are saddle atoms. Here and in what follows all $2$-atoms are assumed to be orientable. It is convenient to code $2$-atoms taking account of the direction of growth of the function $f$ on the $2$-atom by means of $f$-graphs introduced by Oshemkov [72]. The orientable two-dimensional surface $P^2$ of an atom is symplectic, that is, the vector field $v=\operatorname{sgrad}f$ defines a Hamiltonian system with one degree of freedom on this surface (and $f$ is a first integral of this system). Recall the construction of $f$-graphs for orientable atoms (otherwise some edges of the graphs are marked by $+1$ or $-1$). Consider the boundary level $f=c-\varepsilon$ on a saddle $2$-atom. It consists of several circles, and a small neighbourhood of each saddle point intersects the union of these circles in two open arcs. Fix a point on each arc and associate these points with vertices of the graph. Each vertex has degree $3$: two oriented edges (an incoming and an outgoing one) and one non-oriented edge are incident to it. Oriented edges of the graph correspond to shifts along the Hamiltonian field $v=\operatorname{sgrad}\,f$ that take one distinguished point to another along circles of the level $f=c-\varepsilon$. The non-oriented edge connects points on the two arcs in a neighbourhood of the same saddle point (we will assume that oriented and non-oriented edges are coloured with different colours). Note that we can code the $f$-graph of an atom with complexity $k$ by a pair of permutations from $S_{2k}$, one of which is formed by $k$ independent transpositions (defining a correspondence between pairs of vertices of the graph and saddle points of the $2$-atom), while cycles forming the second are defined by shifts induces by the Hamiltonian field along circles on the level $f=c-\varepsilon$ (taking one distinguished point to another). This representation turns out to be convenient for modelling $2$- and $3$-atoms by billiard books [36]. Remark 1. Te classification of singularities of systems with one degree of freedom was significantly extended by Nikolaenko [73], who also generalized their codings by $f$-graphs in the process. For an orientable Morse $2$-atom a non-oriented edge of the $f$-graph is now interpreted as a cycle of length two coloured differently from the oriented edges of the $f$-graph. To a degenerate singularity of multisaddle type (a singular point with $2k$ incoming separatrices) there corresponds a tuple of $k$ vertices of the graph ordered cyclically. It defines $k$ edges oriented compatibly with the cycle and coloured differently from the edges connecting vertices of two distinct saddles. Recently Kuznetsova realized such atoms by means of integrable billiard books. Remark 2. Subsequently, Oshemkov extended the construction of an $f$-graph to the case of non-degenerate saddle singularities of rank $0$ of integrable systems with $m$ degrees of freedom. A neighbourhood of each point is leafwise homeomorphic to the product of $m$ neighbourhoods of a Morse saddle by $X^2$, and the singularity is defined by a set of $m$ products of transpositions and $m$ permutations satisfying certain commutation conditions. Using this construction Kibkalo could realize semilocal non-degenerate saddle singularities of rank 0 by means of billiards with repulsive Hooke potentials. 1.3. Bott singularities of integrable systems. $3$-atoms. Fomenko’s theoremConsider an integrable Hamiltonian system with two degrees of freedom $v=\operatorname{sgrad}H$ on $M^4$, with Hamiltonian $H$ and additional integral $f$. We restrict $v$ and $f$ to a $3$-manifold $Q^3_h$. Let $L$ be a singular leaf of the Liouville foliation on $Q^3_h$. Definition 7. By a $3$-atom we mean a three-dimensional invariant neighbourhood $U(L)$ of the singular leaf $L$, which is foliated by level surfaces of the integral $f$ and which is considered up to leafwise equivalence. If $f$ is a Morse function, then this $3$-atom is called a Bott $3$-atom. On critical circles of $f$ orientation is defined by the flow $v=\operatorname{sgrad}H$. The neighbourhood $U(L)$ is always orientable. The leafwise equivalence must preserve the orientation of critical circles. Theorem 2 (bijection between the Morse $2$-atoms and Bott $3$-atoms; Fomenko [6]–[8]). 1) The 3-manifold $U(L)$ is a Seifert manifold carrying a Seifert fibration with circles as fibres and two-dimensional base $P(L)$. Singular fibres of this fibration (if exist) have the same type $(2,1)$. 2) These singular fibres are precisely critical circles of the first integral $f$ with non-orientable separatrix diagrams. 3) If this Seifert fibration has no singular fibres, then $U(L)$ is the Cartesian product $P(L)\times S^1$, where $P(L)$ is a bordered orientable two-dimensional surface. 4) In the general case the structure of a Seifert fibration on $U(L)$ and the structure of a Lioufille foliation there are compatible in the following sense: each fibre (circle) of the Seifert fibration lies on some leaf of the Liouville foliation. 5) In particular, $f$ is constant on fibres of the Seifert foliation and therefore can be regarded as a Morse function on the base $P(L)$. If there are no singular fibres of type $(2,1)$, then $P(L)$ is a $2$-atom with function $f$ as described above. The case of singular fibres of type $(2,1)$ is described below. Namely, if the level $L$ contains a critical circle with non-orientable separatrix diagrams (that is, of type $(2,1)$), then the Seifert fibration has a ‘section’ $\widetilde{P}$ in $U(L)$ with the following properties:
Claim 1 ([7], [8]). 1) The map $g$ is an involution on $\widetilde{P}$. Its fixed points are precisely the points of intersection of the ‘section’ $\widetilde{P}$ with singular points (circles) of the Seifert fibration. 2) The base $P(L)$ of the Seifer fibration on a $3$-atom $U(L)$ is the quotient of the surface $\widetilde{P}$ by the action of $g$. Definition 8. We call the base $P(L)$ mentioned above a ‘$2$-atom with stars’, where ‘stars’ correspond to fixed points of the involution $g$, that is, to singular fibres of type $(2,1)$ of the Seifet fibration. Thus, Fomenko’s theorem states that there is a bijection beween the $3$-atoms and the $2$-atoms (with and without stars). In the case of a $2$-atom without star vertices the $3$-atoms of the Cartesian product of this $2$-atom by a circle. $2$-atoms with stars are the bases of Seifert fibrations with singular fibres on $3$-atoms, and, we repeat that the star vertices are in bijection with the singular fibres of type $(2,1)$ of the Seifert fibration. Since $3$-atoms describe bifurcations of Liouville tori as they go over a critical level of the integral $f$ on $Q^3_h$, they are uniquely classified by $2$-atoms (with and without stars). 1.4. Topology of isoenergy surfaces $Q^3$ of integrable systemsDefinition 9. Let $(H)$ denote the class of smooth orientable compact closed (without boundary) $3$-manifolds that are isoenergy surfaces of non-degenerate (Bott) integrable systems, that is, systems integrable by Bott integrals. Now consider two $3$-manifolds: a solid torus $A$ and the certesian product of a $2$-disc with two holes by a circle, denoted by $B$. Definition 10. Let $(Q)$ denote the class of orientable closed compact $3$-manifolds representable in the form $Q^3=aA+bB$, where $a$ and $b$ are non-negative integers and ‘+’ denotes the gluing of two manifolds along diffeomorphisms of boundary tori. Definition 11. Let $({\rm Wa})$ denote the class of Waldhausen (or graph) manifolds, that is, orientable compact closed $3$-manifolds such that (a) the manifold contains a finite set of disjoint tori; (b) after removing these tori one obtains an open $3$-manifold each connected component of which is a Seifert fibration with fibres equal to circles over a 2-manifold (which may have a boundary and is not necessarily orientable). Definition 12. Let $(H')$ denote the class of orientable compact closed $3$-manifolds that are three-dimensional isoenergy surfaces of Hamiltonian systems integrable by tame integrals $f$. A smooth first integral is said to be tame if for each critical level of it there exists a homeomorphism of the whole $3$-manifold that takes this level to a polyhedron. Theorem 3 (Brailov, Matveev, Fomenko, and Zieschang [6]). 1) The four classes introduced above coincide: 2) The class $(H)$ does not exhaust the class of smooth connected orientable compact $3$-manifolds. 3) For any two manifolds in $(H)$ their connected sum also belongs to $(H)$. 4) If a manifold in the class $(H)$ can be represented as a connected sum of two manifolds other than spheres, then both these manifolds also belong to $(H)$. 1.5. Topological invariants of the Liouville foliations on non-singular surfaces $Q^3$Consider the base of the Liouville foliation on a non-singular surface $Q^3_h$, that is, on a surface where $dH \ne 0$. This is a one-dimensional graph $W$ whose edges correspond to one-parameter families of regular Liouville tori and whose ‘vertices’ correspond to bifurcations of these tori. To each ‘vertex’ we assign a symbol denoting the corresponding $3$-atom. Definition 13. The resulting graph $W$ with atom vertices is called a rough molecule $W$ or a Fomenko invariant. The Fomenko invariant is a classifying invariant for rough Liouville equivalence. Invariants of rough Liouville equivalence on an isoenergy manifold $Q^3$ and a symplectic manifold $M^4$ (away from a number of fibres) are the subject of [3], [74], and [75] (also see [76]). The molecule $W$ contains much information about the Liouville foliation, but this information is not sufficient for the classification of foliations up to Liouville equivalence. Let us cut each edge of a molecule at the midpoint. Then the molecule falls into separate $3$-atoms. If we want to glue them back, then the graph $W$ provides information about the pairs of boundary tori that must be glued together. To describe such a gluing, for each cut edge it suffices to indicate the gluing matrix $C$ describing the isomorphism between the fundamental groups of the tori to be glued. To do this, on each torus we must fix a coordinate system, that is, a pair of independent oriented cycles which are generators of the fundamental group $\mathbb{Z}\oplus\mathbb{Z}$ of the torus. Such a coordinate system $(\lambda,\mu)$ (which we call admissible) can be defined on each $3$-atom using its geometry. For a detailed description, see [5]–[8] and [77]. Now consider an arbitrary edge $e_i$ of the molecule $W$ and choose an orientation on it, for instance, in the increasing direction of the function $f$. We have cut this edge along a Liouville torus and have defined admissible coordinate systems $K_1$ and $K_2$ on the sides of the cut. Regarding these pairs of cycles as bases in the one-dimensional homology of the torus we obtain an integer gluing matrix $C_i$ of size $2\times 2$ with determinant $-1$. Although these matrices are not uniquely defined, this does not affect the constructions below. From the set of these matrices we construct invariants (not depending on the freedom in the choice of admissible matrices any longer). Namely, these are rational marks $r_i$ and $\varepsilon_i=\pm1$ on the edges $e_i$ and integer marks $n_k$ on so-called families. Families are defined as follows. By an infinite edge of the molecule we mean the edge with mark $r_i$ equal to infinity. Other edges are called finite. We cut the molecule across all finite edges. Then it falls into several connected components. Those not containing minimax atoms $A$ (see above) are called families. Such $3$-atoms $A$ are diffeomorphic to a solid torus. If all edges of a molecule are finite, then each of its saddle atoms is a family by definition. Definition 14. A molecule $W$ endowed with numerical marks $r_i$, $\varepsilon_i$, and $n_k$ is called a marked molecule or a Fomenko–Zieschang invariant. One of the central results in the theory of Liouville classification of integrable systems with two degrees of freedom is as follows. Theorem 4 (Fomenko and Zieschang; see [4] and [77]). Two integrable systems $(v,Q)$ and $(v',Q')$ on isoenergy $3$-surfaces $Q$ and $Q'$ are Liouville equivalent if and only if their marked molecules $W^*$ and $W^{*\prime}$ coincide. In the construction of the marked molecule $W^*$ we used the orientations of the manifold $Q$, the critical circles of the integral $f$, and the edges of the molecule. Changing these orientations changes the molecule in general. For details, see [4] and [6]. On the other hand two marked molecules obtained one from the other by changing the orientation of several edges are set to be coinciding. Theorem 5 (realization theorem; Bolsinov and Fomenko [5]). Each marked molecule defined abstractly can be realized as the marked molecule of some non- degenerate smooth integrable Hamiltonian system. Thus the Fomenko–Zieschang discrete invariant classifies (up to Liouville equivalence) all non-degenerate integrable systems with two degrees of freedom on isoenergy $3$-manifolds. In 1990 Fomenko stated the programme of creating a fundamental Atlas of Integrable Systems in which, on the basis of the calculation of marked molecules, the main series of systems with two degrees of freeom currently known in physics, mechanics, geometry, and topology, would be classified in the sense of Liouville. Apart from the classification of systems on isoenergy $3$-surfaces, it is very important for such an Atlas to classify four-dimensional singularities of systems on a phase manifold $M^4$ (here we mention results due to Zung [12] and Oshemkov [13]). This programme has been implemented to a significant extent and is currently carried out by such authors as Fomenko, Bolsinov, Oshemkov, Matveev, Kudryavtseva, Vedyushkina (Fokicheva), Konyaev, Kinkalo, Fedoseev, and also Anoshkina, Antonov, Belozerov, A. Brailov, Yu. Brailov, Draović, Dullin, Zav’yalov, Ivochkin, Karginova, Zhila, Kalashnikov (jr), Kantonistova, Kobtsev, I. Kozlov, Korovina, Kruglikov, Kuztetsova, Lepskii, Morozov, A. Moskvin, V. Moskvin, Zung, Nikolaenko, Novikov, Orel, Pogosyan, Polyakova, Pustovoitov, Radnovich, Ryabov, Selivanova, Sidel’nikov, Skvortsov, Slavina, Sokolov, Solodskikh, Takahashi, Timonina, Topalov, Tuzhilin, Haghighatdoost, Kharlamov, Kharcheva, Khorshidi, and Khotin. A long list of publications on this subject was presented in books by Fomenko [78] and Bolsinov and Fomenko [6], and in papers by Fomenko and Vedyushkina [55], [79], and Fomenko and Kibkalo [34], [66]. As a result, remarkable pairs of well-known integrable systems considered previously significantly different from one another turned out to be Liouville equivalent. In other words, it turned out that, in spite of the different nature of these systems, the closures of almost all integral trajectories of these systems are the same, that is, they possess the same Liouville foliations. For instance, the Jacobi problem (the geodesic flow on an ellipsoid) turned out unexpectedly to be Liouville equivalent and even continuously oribital equivalent (but not smoothly orbital equivalent) to the Euler top in rigid body dynamics (the Fomenko–Bolsinov theorem: see [6] and [11]). Various authors have calculated the Fomenko–Zieschang invariants for many systems and thus have obtained their Liouville classification. Here are some examples which have already been included in the Atlas. These are some classical integrable cases in the dynamics of heavy rigid bodies, namely, the Euler, Lagrange, and Kovalevskaya cases [56], [80], the Joukowsky case [80], the Clebsch case [81], the Steklov and Sokolov cases [82], and the Goryachev–Chaplygin–Sretenskii case. In addition, some of these systems have been Liouville classified in the case when gyrostats are added [83], as have also been some of their non-holonomic analogues (see, for instance, Zhila’s investigation of the problem ‘Chaplygin ball with rotor on a rough surface’). Linearly and square integrable geodesic flows on two-dimensional surfaces (a sphere, a projective plane, a torus, a Klein bottle) have also been Liouville classified: see details in § 3.4. The same can be said about integrable geodesic flows on 2-surfaces of revolution with a potential or a magnetic field added [84]–[87]. A Liouville classification of multiparameter analogues of the Kovalevskaya system on Lie algebras was obtained [88]–[93]. Also, pseudo-Euclidean analogues of mechanical systems introduced in [94] and, in particular, an analogue of the Kovalevskaya system [95] are under investigation. 2. Integrable billiardsBy a mathematical billiard one usually means a point mass moving without loss of velocity along a plane domain $\Omega \subset \mathbb{R}^2(x,y)$ bounded by a smooth or piecewise smooth curve $\gamma=\partial \Omega$ (the edge of the billiard table). Since the metric is Euclidean, covectors can be identified with vectors — and in what follows we view $M^4=T\Omega/\sim$ as the phase space for transparency. Elements of $T\Omega$ are pairs of a point $P=(x,y)$ and a vector $\vec{v}=(v_x,v_y) \in T_P\Omega$. The Hamiltonian (energy) of the system is equal to $H=|\vec{v}|^2=v_x^2+v_y^2$. The identification of some ‘distant’ pairs point-vector in $T\Omega$ is based on the presence of reflection. Given a point $P \in \gamma=\partial \Omega$ we identify two vectors $\vec{v}$ and $\vec{v}'$ is they have the same moduli and their difference is orthogonal to the tangent line to $\gamma$ at $P$. This corresponds to perfectly elastic reflection, when the incidence and reflection angles are equal. If $\gamma$ makes no corner at $P$ and $\vec{v}$ is transversal to $\gamma$, then we identify it with a unique vector $\vec{v}'$ (the tangent vector to $\gamma$ is the limit position of pairs of such identified vectors). If $P$ is a corner point on $\gamma$, then it is usually assumed that the internal angle is $\pi/2$. If $\vec{v}$ is transversal to both smooth arcs of $\gamma$, then the pair $(P,\vec{v})$ is identified with three other pairs point-vector, while if $\vec{v}$ is tangent to one of the arcs, then this pair is identified with just one other pair. Thus, for the planar billiard in $\Omega$ the preimage of a point $P \in \Omega$ in a non-singular energy level $Q^3_h \subset M^4\colon H=h$ with respect to the projection $\pi\colon (P,\vec{v}) \to P$ is homeomorphic to a circle when $P \in \operatorname{Int}\Omega$ and to a line segment when $P \in \gamma$. Occurring an a corner point with opening angle $\pi/2$ can be interpreted as a double reflection of the particle, from each of the boundary arcs. Some other dynamical systems involving bouncing off and reflection are closely related to such systems, for instance, outer billiards [96], [97], systems in the theory of mirrors in complicated domains [98], systems in the invisibility problem [99], ordered billiard games [100]. Note here the recent paper [101] by Frǎczek and Rom-Kedar. The question of the ergodicity of billiard systems and the properties of their trajectories often attracts much interest. 2.1. Integrability of billiard systems and Birkhoff’s conjectureThere are different definitions of the integrability of a billiard system (see, for instance, [21]). We will be based on Liouville integrability, namely, assume the existence of a first integral that is involutive with and functionally independent of the energy $H$. Everywhere away from the preimage of the boundary the phase space inherits the smooth and symplectic structures from the contangent bundle, so that involutivity is well defined. In considering reflections of trajectories from the boundary we assume that the first integral is continuous. This concept of ‘piecewise smooth’ Liouville integrability of billiard systems was proposed by Fomenko and described in [30]. The simplest examples of integrable billiard systems are billiards inside a disc, a rectangle, and an ellipse. In the first case preserved is the radius of the circle tangent to all links of the polygonal trajectory, and in the second preserved is the non-oriented angle between a fixed line and the straight lines containing links of the trajectory. Birkhoff [26] regarded a billiard inside an ellipse as a limiting case of the Jacobi problem of the motion along geodesic lines on an ellipsoid $E^2 \subset \mathbb{R}^3$ as the semiminor axis tends to zero. Then umbilic points on the ellipsoid turn to the foci of the boundary ellipse. By the Jacobi–Chasles theorem the tangents to a fixed geodesic on $E^2$ are tangent to a certain hyperboloid confocal with $E^2$, the same for all points on this geodesic. If this is a hyperboloid of one sheet, then links of the image of the geodesic are tangent to an ellipse, and if it is of two sheets, then they are tangent to a hyperbola. Resulting quadrics have the same foci as the boundary ellipse and belong to the family (in the $Oxy$-plane)
$$
\begin{equation}
(b-\lambda)x^2+(a-\lambda)y^2=(b-\lambda)(a-\lambda).
\end{equation}
\tag{2.1}
$$
Here the parameters $a$ and $b$, $0<b<a$, are the squares of the lengths of semiaxes of the boundary ellipse. The semi-major axis lies on the $Ox$-axis and contains the foci (we will occasionally call the $Ox$-axis the focal axis) and the semi-minor axis lies on the $Oy$-axis. These axes belong to the family (2.1) for $\lambda=b$ and $\lambda=a$ respectively. The value of $\lambda$ is a first integral: the quadric with this parameter is a caustic for all links. The integral can explicitly be expressed in terms of $x$, $y$, $v_x$, and $v_y$ as
$$
\begin{equation}
\Lambda=\frac{-(xv_y-yv_x)^2+bv_x^2+av_y^2}{v_x^2+v_y^2}\,.
\end{equation}
\tag{2.2}
$$
Let $\lambda=0$; then the point $P$ lies on the boundary ellipse and the vector $\vec{v}$ is tangent to the ellipse. if $\lambda=a$, then $P \in Oy \cap \Omega$ and $\vec{v} \parallel Oy$. These levels correspond to the one-dimensional levels $\Lambda=\lambda$ in $Q^3_h$. The level $\lambda=b$ is two-dimensional. Edges of each trajectory lie on straight lines passing through one focus or the other. Singular trajectories on the level $\Lambda=b$ consists of the pairs $(P,\vec{v})$ such that $P\in Ox \cap \Omega$ and $\vec{v} \parallel Ox$. Since confocal quadrics intersect under right angles, $\Lambda$ is preserved [61] under reflections of a trajectory from any curve in the family (2.1). Here the absence of internal angles of $3\pi/2$ ensures that the motion of the billiard ball in the plane domain bounded by arcs of confocal quadrics is continuous. By an elementary (confocal) billiard table we mean a compact connected part of the plane with boundary formed by arcs of conformal quadrics in the family (2.1) and not containing angles of $3\pi/2$. The equivalence relation between such tables introduced by Vedyushkina in [31] (which is called structure equivalence in what follows) preserves the foliation and, omitting some details, can be defined as follows. Two billiards are equivalent if their table domains can be taken one to the other by an isometry of the plane or if there exists a continuous deformation of the parameters $\lambda_i$ of the boundary arcs of one table to the parameters of the other that preserves the type of each boundary arc (elliptic for $\lambda_i\in (-\infty,b)$, focal for $\lambda_i=b$, and hyperbolic for $\lambda_i \in (b,a]$), and the homeomorphism class of the table $\Omega$. Examples of elementary billiards are shown in Fig. 1 (the notation agrees with [31]). The idea of Birkhoff’s billiard conjecture (in various versions) consists in finding a criterion for the integrability of a billiard system. One statement is as follows: is it true that a billiard inside a smooth plane curve is integrable only when this curve is an ellipse? In the context of the polynomial version of this conjecture (that is, the question on the existence of a first integral which is polynomial in momenta) important results were established by Bolotin [17]. For a domain with piecewise smooth closed boundary he showed that each smooth arc must lie on an algebraic curve whose algebraic closure in $\mathbb{C}P^2$) either has degree $1$ or $2$ or contains singular points. In particular, if the edge of the table is smooth then either it is an ellipse or the dual curve to it contains singular points. The next step was done by Bialy and Mironov [19] (who used the construction of an angular billiard introduced in [19]): if a smooth arc on the boundary is not a line segment, then the dual curve in $\mathbb{C}P^2$ has degree $2$ or contains singular points; moreover, all its singular and inflection points lie on the union of two isotropic lines. There are also examples of billiard tables in [19] for which polynomial non-integrability follows from this result but does not follow from the preceding results. The above results have generalizations to surfaces of constant curvature (see [18] and [20], and also the more special paper [102]). In [21] and [22] Glutsyuk completed the proof of the polynomial version of Birkhoff’s conjecture. Assume that the piecewise smooth boundary of a compact simply connected domain in a constant curvature space contains at least one segment not lying on a geodesic. Then the billiard in this domain is integrable if and only if the boundary consists of arcs of confocal quadrics and maybe also arcs of some admissible geodesics (in the elliptico-hyperbolic case these can be intervals of the principal axes of the family). Other versions of Birkhoff’s conjecture and close problems are also under intensively investigation. Kaloshin and Sorrentino [23], [24] proved a local version of Birkhoff’s conjecture: a small perturbation of an ellipse in the plane such that the billiard inside it is integrable is itself an ellipse. Note also the earlier paper by Tabanov [103], where conditions for the absence of analytic integrability of billiards were discussed for domains close to ellipses (and obtained by global symmetric analytic perturbations of an ellipse). The question of the integrability of billiards in multidimensional domains was also considered by Kozlov [104] and Glutsyuk [105] in recent papers. Many interesting problems also arise in the analysis of the dynamics of (integrable, in one sense or another) billiards. For example, Treschev [106] asked whether or not a rigid rotation through some angle is conjugate to the billiard map for a suitable symmetric billiard defined in a small neighbourhood of a periodic (elliptic) trajectory of length $2$. The function defining locally the surface bounding the billiard table in a neighbourhood of points of reflection of this trajectory is still unknown. It was also shown in that paper that for angles incommensurable with $\pi$ there is a solution given by a formal series and numerical modelling was carried out. In the subsequent papers [107] and [108] numerical and analytic investigation of this problem was continued for billiards with one-dimensional boundary and for higher-dimensional cases. The ‘particular’ solutions obtained are different from quadrics. 2.2. Integrable billiards on piecewise flat manifolds and CW-complexes with permutationsAs pointed out above, the class of integrable billiards on planar tables with curvilinear boundaries (in the absence of potentials and magnetic fields) reduces to the class of confocal billiards on tables bounded by arcs of confocal quadrics without internal angles of $3\pi/2$ on the boundary and the class of circular billiards, which are the degenerations of confocal billiards as the foci of quadrics tend to their centres (such billiard tables are bounded by arcs of concentric circles and segments of their radii so that the angles at corner points are $\pi/2$). Although this class is quite narrow (for example, from the standpoint of the topology of Liouville foliations), it admits the following cardinal extensions proposed by Vedyushkina. Definition 15. A generalized (topological) billiard is a dynamical system on two dimensional oriented compact manifold glued (by means of isometries) of elementary confocal billiard sheets: Remark 3. For a topological billiard its projection onto the plane is well defined; it is an isometry and a homeomorphism when restricted to the closure of each planar sheet. Motion along a topological billiard is defined as follows: motion inside a planar billiard sheet is standard, along straight line segments, with natural reflections at the boundary. As soon as the point mass occurs on a gluing edge. it is reflected and continued its motion along another billiard sheet. The introduction of topological billiards expanded considerably the class of billiards under investigation. In [31] and [33] they were classified both structurally and topologically: the Fomenko–Zieschang invariant were calculated for all of them. A significantly larger extension was the class of billiard books introduced by Vedyushkina [32], [36]. These tables are CW-complexes with edges ($1$-cells) endowed by permutations of the set of incident $2$-cells (see details in [32]). Consider a two-dimensional CW-complex with two-dimensional cells that are planar billiards bounded by arcs of confocal quadrics. One-dimensional arcs of this complex are parts of boundaries of elementary billiards, namely, arcs between corners on boundary curves. We number all two-dimensional cells, and to each one-dimensional edge of the complex (a ‘spine’ of the book) we assign a cyclic permutation of the indices of sheets adjoining this edge. Let us project all billiard sheets isometrically onto the plane. If several edges of the CW-complex project onto the same arc on the plane, then we combine the cyclic permutations assigned to these edges into a single permutations (the cycles are obviously independent). To have a continuous motion of the particle along the book we need that the permutations at zero-dimensional cells commute. In terms of the projection this means that the permutations assigned to arcs of two quadrics in a neighbourhood of a point of intersection of these quadrics commute (see Fig. 2). We call this two-dimensional complex with assigned permutations a billiard book. Billiard motion along a book is defined as follows. Inside two- dimensional cells the motion is as usual. Assume that after the motion along the sheet with index $i$ the points mass occurs on a spine of the book. Then, after bouncing off this spine it continues its motion along the sheet $\sigma(i)$. If the sheets with indices $i$ and $\sigma(i)$ lie on the same side of the spine, then bouncing off occurs with reflection, and if they lie on distinct sides, then the point is not reflected, but goes ‘through’ the spine (see Fig. 2). Remark 4. The condition that at corners of the book permutations commute is necessary and sufficient for a consistent definition of the extension of a trajectory occurring at the vertex of a corner. Let $l_1$ and $l_2$ be arcs of quadrics that have a common point $O$, and let $\sigma_1$ and $\sigma_2$ be the commuting permutations assigned to these arcs. A trajectory occurring at $O$ is on the one hand the limit of close trajectories that first bounce off the spine $l_1$ and then off the spine $l_2$. Such trajectories change the index of the billiard table in accordance with the permutation $\sigma_2\circ\sigma_1$. On the other hand the trajectory occurring at $O$ is the limit of trajectories bouncing the spines in the other order, and it changes sheets in accordance with the permutation $\sigma_1\circ\sigma_2$. Thus, the point mass occurring at the corner changes the sheet in accordance with the permutation $\sigma_1\circ\sigma_2=\sigma_2\circ\sigma_1$ because $\sigma_1$ and $\sigma_2$ commute. The resulting class is quite large, and there is no full classification for billiards in this class at the moment. The problem reduces to the description of sets of permutations possessing certain properties. A number of recent advances here are due to Vedyushkina and Kibkalo [48]. In that paper they considered billiard books of low complexity, namely, books such that the projections of spines of the book onto the plane can have just two images. For some billiard books the Fomenko–Zieschang invariants of the arising Liouville foliations were calculated. Another approach to the classification of billiard books was proposed in the recent paper [34] by Kibkalo and Fomenko. They introduced the operation of bending a book along a quadric. Using this operation one can reduce (in more than one way) the number of quadrics with different parameters $\lambda_i$ onto which $1$-cells of the book are projected. As a result, they obtained nine families of books, each of which is defined by a fixed number of permutations and commutation conditions for permutations. So the problem of the classification of books can be solved by direct search. Note that one and the same book can be reduced to different forms. The following fact holds for the billiard systems on topological billiards and billiard books. The piecewise smooth phase manifold $M^4$ is piecewise symplectic by a theorem of Kharcheva [42]. The system on a billiard book is piecewise Hamiltonian and integrable because there exists an additional integral, the parameter of a confocal quadric (2.2), expressed in terms of the coordinates $(x,y,v_x,v_y)$ of the point and vector in the $Oxy$-plane that are the projections of a pair point-vector on the table complex. As in the case of planar billiards, consider the isoenergy level $Q^3_h \subset M^4\colon H=v_x^2+v_y^2=1$. By Kharcheva’s theorem [42], for each book this isoenergy surface $Q^3$ is a piecewise smooth 3-manifold. There is a, generally speaking, piecewise smooth foliation of $Q^3$ by level surfaces of the first integral $\Lambda$. For many systems it has been verified directly that this foliation contains a finite number of singular leaves and their neighbourhoods are leafwise homeomorphic to Fomenko $3$-atoms, while all non-singular leaves are, as in the smooth case, 2-tori (a piecewise smooth analogue of Liouville’s theorem), and almost all of them are the closures of phase trajectories. Note that the dynamical system of a book table (which is not a two-dimensional piecewise smooth manifold, that is, not a topological billiard) is not invariant with respect to time reversal in the case when the point mass in this system can occur on a $1$-cell with permutations consisting of cycles of length at least $3$. The reason is that the condition $j=\sigma(i)=\sigma(\sigma(j))=\sigma^2(j)$ is not satisfied for gluing edges ($1$-cells of the complex). Remark 5. Other examples of irreversible systems are known in geometry. Thus, billiard books combine two properties, integrability and irreversibility, in an interesting way. When time changes to reverse time the trajectory jumps from one Liouville torus to another, which lies necessarily on the same level of the integral as the original torus. The following analogy is possible here: Bolsinov and Taimanov discovered a similar picture, when a system is integrable and chaotic (non-integrable) at the same time, on different level surfaces (see [109]–[111]), and has a positive topological entropy at the same time. 2.3. Fomenko’s conjecture about billiardsThe introduction of billiards books has not only allowed one to extend significantly the class of integrable billiard systems but has also led to the discovery of new Liouville foliations (be they just piecewise smooth). On the one hand these foliations (coded by Fomenko–Zieschang invariants) were not encountered previously in classical problems of dynamics, but on the other hand the billiard systems corresponding to them can be described geometrically. In this connection Fomenko [35] stated a general conjecture on the realization of arbitrary Liouville foliations (that is, marked molecules) of non- degenerate integrable systems with two degrees of freedom by integrable billiards (in the class of Liouville equivalence). Note that there are certain restrictions on the Liouville foliations to be modelled. First, this is non-degeneracy, that is, we only consider integrable systems on isoenergy $3$-surfaces $H=\operatorname{const}$ on which the Hamiltonian $H$ is non-degenerate: $dH\ne 0$ on the whole of the surface $Q^3$. Second, we assume that the additional integral $f$ is a Bott function on $Q^3$. Third, the systems under consideration are nonresonance, that is, the windings on Liouville tori are non-resonance and dense for almost all values of $f$ and $H$. Conjecture Fomenko. The following objects can be realized in the class of Liouville foliations: The following holds for rotation functions. This conjecture is complemented by its ‘local’ version stated in [37]. It claims that arbitrary values of a numerical mark, or a marked neighbourhood of an element (an edge, a vertex, a family sibgraph) on the Fomenko–Zieschang graph invariant can be realized. Conjecture (F-E) is closely connected with the construction of analogues of Fomenko–Bolsinov orbital invariants for billiards. A number of explicit examples was considered by Vedyushkina [112]. Remark 6. Conjectures A, B, D, and the local version are necessary conditions for C, the main conjecture: the local conjecture and Conjecture A assert that all components of the invariant can be realized, Conjectures B and D assert that the class of integrable billiards and the class of integrable systems coincides up to certain weaker relations than Liouville equivalence, the homeomorphy of bases (B), local lifts (local version), and the homeomorphy of the levels $Q^3$ taking no account of the foliations. Conjectures B and D do not imply one the other. Conjecture A was fully proved by Vedyushkina and Kharcheva [32]. Consider a billiard table in the class $A_0'$ of billiards that is bounded by an arc of an ellipse, the focal line ($Ox$-axis) and two arcs of hyperbolae, one of which is convex. Recall that the $Ox$-axis also belongs to the focal family for $\lambda=b$. Theorem 6 (Vedyushkina and Kharcheva). For each orientable saddle $3$-atom (with or without stars) a billiard book $\Omega$ can algorithmically be glued of several copies of the billiard $A_0'$ (see Figure 1) so that the Liouville foliation in the preimage $\Lambda^{-1}((b-\varepsilon,b+\varepsilon)) \subset Q^3$ of a neighbourhood of a singular value of the integral is leafwise homeomorphic to this atom. In the case of the atom $A$ a similar result holds for $\Omega=A_0'$ and the level $\lambda=0$. The billiards so constructed model orientable Bott $3$-atoms. Hence the permutation assigned to an interval of the $Ox$-axis is a product of independent transpositions. If some independent cycles have length at least $3$, then the singular level obtained for $\Lambda=b$ belongs to a non-Bott $3$-atom (see an example in Fig. 3). Here is a stronger version of Conjecture A, established already, for non-Bott $3$-atoms. Conjecture $\boldsymbol{\widetilde{{\rm A}}}$. In the class of Liouville foliations for integrable billiards not only Bott bifurcations of Liouville tori can be realized, but also rather rich classes of bifurcations of Liouville tori described by ‘non-Bott’ $3$-atoms, including multisaddle singularities of rank $1$. For example, of great interest are bifurcations of tori described by Seifert fibrations whose bases are not necessarily Morse 2-atoms. In this case the Seifert fibration can have arbitrary singular fibres of type $(p,q)$. The first steps in the realization of such non-Bott singularities by integrable billiards were made by Kuznetsova. She classified bifurcations of Liouville tori arising in the Liouville foliations for billiard books glued of three $A_0'$ sheets. In particular, she realized non-Bott $3$-atoms corresponding to non-Morse $2$-atoms with one vertex by billiards (see an example of a non-Morse $2$-atoms of multiplicity $3$ in Fig. 3, (b)). Conjecture B has also been proved [40]. The proof uses billiard tables $B_0$ disjoint from the focal line and bounded by two arcs if ellipses and two arcs of hyperbolae. One arc of a hyperbola bounding the billiard domain is convex with respecvt to this domain (see Fig. 1). Theorem 7 (Vedyushkina and Kharcheva). Fomenko’s Conjecture B is true: for each rough molecule (Fomenko invariant, a graph containing the types of bifurcation atoms at its vertices) one construct algorithmically a billiard book glued of simplest $B_0$ billiards such that the Fomenko invariant of the corresponding foliation coincides with the prescribed one. In other words, the set of classes of rough Liouville equivalence is realized by billiards. As concerns Conjecture C, an obstruction to the realization of Hamiltonian systems by billiard books (or some subclasses of billiard books, of course) has been found. Theorem 8. The Liouville foliation on $S^1\times S^2$ that has the Fomenko–Zieschang invariant $A$–$A$ with marks $r=\infty$ and $\varepsilon=-1$, that is, the one corresponding to a modified (‘twisted’) Lagrange top [113], cannot be realized as the Liouville foliation on an isoenergy surface of some integrable confocal billiard book. Adding external forces to the billiard system on a table CW-complex $X$ (for instance, adding a magnetic field whose induction is orthogonal to each sheet, permanent in time, and constant with respect to points in $X$) extends additionally the class of realizable foliations. In this new class of magnetic billiards (which we discuss in greater detail in § 4.4) the invariant indicated in the above theorem can be realized. Proposition 1. The Liouville foliation for a ‘twisted’ Lagrange top in a non- singular energy range where $Q^3 \simeq S^1\times S^2$ and the molecule is $A$–$A$ with marks $r=\infty$ and $\varepsilon=-1$ can be realized by the Liouville foliation on an isoenergy surface $R=\operatorname{const} < r_0$ for the magnetic billiard in the annulus between two concentric circles. Since the question of the validity of Conjecture C as a whole is apparently quite complicated, Fomenko distinguished a local version of this conjecture, which consists of six parts [37]. Local Conjecture C (realization of numerical invariants of integrable systems). $\mathrm{C}_{loc}$-1. Let $\gamma$ be an edge with marks $r$ and $\varepsilon$ of a marked molecule $W^{*}$. Then there exists an integrable billiard realizing the combination of marks $r$ and $\varepsilon$ on an edge of its marked molecule. Note that following four cases can occur: the mark $r=p/q$ is finite and $\varepsilon=\pm 1$; $r$ is equal to $\infty$ and $\varepsilon=\pm 1$. $\mathrm{C}_{loc}$-2 (refinement of Conjecture $\mathrm{C}_{loc}$-1). Under the assumptions of Conjecture $\mathrm{C}_{loc}$-1 there exists a billiard realizing an arbitrary pair of marks $r$, $\varepsilon$ on the edge between any two prescribed atoms. $\mathrm{C}_{loc}$-3. Let $S$ be a family with integer mark $n$ in the marked molecule $W^{*}$ of an integrable system. Then there exists an integrable billiard realizing a family with this integer mark $n$. $\mathrm{C}_{loc}$-4 (refinement of Conjecture $\mathrm{C}_{loc}$-3). Under the assumptions of Conjecture $\mathrm{C}_{loc}$-2 there exists a billiard that, apart from a prescribed mark $n$, also realizes the whole family, that is, the graph with prescribed atoms and prescribed set of edges. $\mathrm{C}_{loc}$-5 (realization of a marked neighbourhood of a family). Let $S$ be a family with integer mark $n$ in a marked molecule, and assume that the outer edges $\gamma_i$ carry some arbitrary marks $r_i$ and $\varepsilon_i$. Then there exists a billiard realizing this marked subgraph in its marked molecule. $\mathrm{C}_{loc}$-6 (realization of a marked neighbourhood of an edge). Let $S_1$ and $S_2$ be two families with integer marks $n_1$ and $n_2$ in a marked molecule such that two selected boundary tori in these families are joined by an edge endowed by an arbitrary pair of marks $(r,\varepsilon)$. Then there exists a billiard realizing this marked subgraph in its marked molecule. Theorem 9 (Vedyushkina [38]). Fomenko’s Conjecture $\mathrm{C}_{loc}$-1 hold for each pair of numerical invariants $r$, $\varepsilon$, namely, for each edge of a marked molecule $W^*$ with this pair of marks there exists a billiard such that its marked molecule contains an edge with this pair of marks. Theorem 10 (Vedyushkina [38]). Fomenko’s Conjecture $\mathrm{C}_{loc}$-2 holds in the cases listed in Table 1. More precisely, in seven cases all pairs $r$, $\varepsilon$ of numerical marks can be realized by integrable billiards for edges joining arbitrary prescribed atoms. In the four remaining cases any combinations of marks can only be realized at the current moment for edges connecting certain particular atoms in the series $B$ and $C$. Table 1.Combinations of marks $r$ and $\varepsilon$ on edges of marked molecules of integrable billiards
Remark 7. If the orientation of the isoenergy surface $Q^3$ changes, then admissible coordinate systems change too. Hence the marks on edges change in accordance with the rules below (see [6]). 1) Let the edge connect atoms of the same type, that is, $A$ with $A$, or a saddle with a saddle. Then on a finite edge (when $\beta\ne 0$) the marks $r$ and $\varepsilon$ change signs. On an infinite edge (when $\beta=0$) $r$ and $\varepsilon$ remain the same. 2) Assume that the edge connecting atoms of different types, that is, $A$ with a saddle. Then on a finite edge the mark $r$ changes sign, while $\varepsilon$ does not change. On an infinite edge, conversely, $r$ does not change (remains equal to infinity), while $\varepsilon$ changes sign. Examples of billiard books realizing various cases of pairs of marks $(r,\varepsilon)$ in the Fomenko–Zieshang invariants of integrable billiards are shown in Figure 4. In Table 1 the case of an edge with non-integer mark $r$ which connects an atom $A$ with an arbitrary saddle atom is new. Here we present an algorithmic construction of the required book. Let $V$ denote a saddle atom without stars. Consider the billiard book $B_V$ realizing it in accordance with the Vedyushkina–Kharcheva algorithm. Fix a convex elliptic spine in this book, with a permutation of length $l$ assigned to it. Also fix positive integers $n$ and $k$ ($k<n$). Take $n$ copies of the book $B_V$ and glue them into a single book $B$ by adding the following gluings along convex elliptic and hyperbolic spines. To each copy of $B_V$ we assign an index and call it a ‘chapter’ for convenience. First we glue all chapters $B_V$ one with another along the convex hyperbolic spines. To the new resulting spine we assign a cyclic permutation $\sigma$ of length $n$ which rearranges the indices of chapters (but does not changes the indices of sheets in a chapter). Second, we glue all spines $s_e$ one with another and replace the cyclic permutations of length $l$ assigned to them by the following permutation of length $ln$. After traversing all sheets in a cycle of length $l$ the particle goes over to the first sheet again, but changes the chapter in accordance with the permutation $\sigma^k$. These gluings do not change the form of the atom and the molecule as a whole. However, a mark $r$ equal to $k/n$ arises on the bottom edge connecting $V$ with the atom $A$ corresponding to the spine $s_e$. Using this construction one can prove the following result. Proposition 2 (Vedyushkina). Let $V$ be an arbitrary saddle atoms without stars. Fix some tori below the critical level on it. Fix arbitrary marks $r=t_i/n_i$ on the edges corresponding to these tori. Then an integrable billiard glued from the billiards $A_0'$ and realizing the atom under consideration with fixed $r$-marks on bottom edges is constructed algorithmically. A similar construction enables one to produce arbitrary marks for tori lying over the critical level, provided that the billiard domain $B_1'$ is taken in place of $A_0'$. We present a full proof in a separate paper. Now we go over to the question of the realization of the integer marks $n$ assigned to certain subgraphs of the (Fomenko–Zieschang) invariant which are called families. Theorem 11 (Vedyushkina and Kibkalo [39]). Part $\mathrm{C}_{loc}$-3 of Fomenko’s local conjecture is valid, namely, for each $k \in \mathbb{Z}$ a billiard $\Omega_k$ can algorithmically be constructed for which the Liouville foliation on a non-singular isoenergy surface contains a family with prescribed mark $n=k$. Recall that an elementary billiard domain bounded by an ellipse in the family (2.1), was denoted by $A_2$ in [31] (see Fig. 1). After cutting it along an arc of a hyperbola from (2.1) two elementary domains of type $A_1$ arise, while after cutting it along both arcs of the hyperbola we obtain a domain of type $A_0$ and two symmetric domains of type $A_1$. The domains $A_i$ contain an interval of the focal line and $i$ foci of the family (2.1). The description of the construction of the billiard tables $\Omega_k$. We take $n$ copies $S_1,\dots,S_n$ of a table of type $A_2$ bounded by an ellipse with parameter $\lambda=0$ in the family (2.1). We cut the table $S_1$ along the arcs of the hyperbola with parameter $\lambda=\lambda_1$, the table $S_n$ (for $n>1$) along the arcs of the hyperbola with parameter $\lambda= \lambda_{n-1}$, and we cut the other tables $S_i$, $2 \leqslant i \leqslant n-1$ (provided that $n >2$) along the arcs of the two hyperbolae with parameters $\lambda=\lambda_{i-1}$ and $\lambda=\lambda_i$. Here $b < \lambda_1 < \dots < \lambda_{n-1} <a$. We present the notation for the resulting domains in Table 2. Note that the billiard table $S_i$ is partitioned into a set of sheets $(a_i,x_i,b_i,y_i,c_i)$, or into a set of sheets $(a_i,b_i,c_i)$. Table 2.Notation for sheets of billiard tables
We construct the billiard table $\Omega_k$ from the above sheets by gluing them along negative and positive branches (lying in the half-planes $x <0$ and $x >0$, respectively) of boundary hyperbolae with permutations $\sigma_i$ and $\rho_i$, respectively. Table 3 contains these permutations, and Fig. 5 shows the billiard table $\Omega_3$. Table 3.The permutations on gluing spines of the tables $\Omega_k$
A more general result also turns out to be valid, which allows one to make a significant advancement towards the proof of Conjecture $\mathrm{C}_{loc}$-4. Theorem 12 (Vedyushkina [48]). Let $W$ be a rough molecule with atoms without stars as vertices such that the atoms $A$ are removed from the pending vertices. Assume that the edges connecting saddle atoms in the graph carry the marks $r=\infty$ and $\varepsilon=1$. Then a billiard book $\mathbb{B}(W,m)$, whose marked molecule looks as follows, can be glued algorithmically of billiard domains $A_0$ and $A_1$. To each free bottom edge of $W$ one assigns an atom $B$ with two edges going out of it and ending at atoms $A$. To each free top edge of $W$ one assigns an atom $A$. Each edge, except the ones between saddle atoms, carries the marks $r=0$ and $\varepsilon=1$. The family formed by $W$ has the mark $n=m$, and all other families have the marks $n=0$. One example of such a book is shown in Fig. 6. The molecule $W$ contains one atom $B$. To obtain an arbitrary molecule of the form indicated in Theorem 12, in place of $b_2$ we must glue in a union of several billiard domains $A_0$, glued along the hyperbolic arcs, with the same permutations as the ones assigned to the elliptic arcs of billiard domains $B_0$ in the Vedyushkina–Kharcheva algorithm for rough molecules. Now we have managed to combine the results established in the proofs of parts $\mathrm{C}_{loc}$-1 and $\mathrm{C}_{loc}$-3. The set of examples obtained is quite interesting in the context of the most general parts of the local conjecture, namely, $\mathrm{C}_{loc}$-5 and $\mathrm{C}_{loc}$-6, which concern the realization of marked neighbourhoods of a family or an edge: the Fomenko–Zieschang invariants obtained involve a pair of families and an edge connecting them such that (i) a family and an outer edge of it carry an arbitrary mark $n=-k$ and a rational mark $r$ with an arbitrary denominator $m$, respectively (part $\mathrm{C}_{loc}$-5); (ii) the outer edges of one o the families carry different ‘non-trivial’ values of the rational mark, namely, $r=2/m$ and $r=-1/m$ (part $\mathrm{C}_{loc}$-5); (iii) two families with non-zero marks $n=-2$ and $n=-k$ are connected by an edge with non-trivial mark $r$ equal to $-1/m$ (part $\mathrm{C}_{loc}$-6); (iv) two families in the same invariant carry different sets of marks $r$ on their outer edges: $r=-2/m$ on the edge between the families, $r=1/m$ on the outer edges of one family, and $r=0$ on the outer edges of the other (several ‘local’ foliations are combined into a single ‘global’ one). Proposition 3. Consider a billiard book glued from $m$ discs bounded by a fixed ellipse (the integer $m$ is greater than $2$). The unique spine is endowed with a cyclic permutation of $m$ elements. Then the Fomenko–Zieschang invariant describing the Liouville foliation of an isoenergy surface of such a book is shown in Fig. 7, (a), for odd $m$ and in Fig. 7, (b), for even $m$. Theorem 13. Consider a billiard book glued of an odd number $m$ of copies of the table $\Omega_k$ along the elliptic boundaries. This Fomenko–Zieschang invariant is as shown in Fig. 8. It contains combinations of marks realizing series of examples to parts $\mathrm{C}_{loc}$-5 and $\mathrm{C}_{loc}$-6 of Fomenko’s local conjecture. Part D of Fomenko’s conjecture has not been established yet. Note that currently in classical integrability cases of rigid body dynamics, in accordance with Smale’s conjecture, an isoenergy surface is a disjoint sum of manifolds of one of the following types. This is a 3-sphere $S^3$, a projective space $\mathbb{R}P^3$, or a connected sum of a finite number of Cartesian products $S^1\times S^2$. In investigations of geodesic flows on two-dimensional surfaces the following manifolds can also be isoenergy surfaces: a 3-torus and a lens space $L(4,1)$. All these $3$-manifolds arise in billiard systems. Moreover, the following result turns out to hold. Theorem 14 (Vedyushkina [46]). Consider a manifold $M$ equal to the connected sum of lens spaces $L(n_1,k_1),\dots,L(n_m,k_m) $ and $l$ copies of the Cartesian product $S^1\times S^2$. Then a billiard book with isoenergy surface homeomorphic to $M$ can be constructed algorithmically. Here is a brief description (see details in Figure 9) of such a (not necessarily integrable) billiard book. Consider the integers $N=\text{GCM}(n_1,\dots,n_m)$ and $g_i=Nk_i/n_i$. Fix a simply connected billiard domain $\Omega$ whose boundary is a quadrilateral $ABCD$. To have an integrable billiard, as this quadrilateral we can take, for instance, a billiard domain $A_0$ bounded by an ellipse and two arcs of a hyperbola. The required book is glued of $N(m+2l)$ copies of $\Omega$. The spines are endowed with the same permutations.
In the general case manifolds homeomorphic to a connected sum of lens spaces and Cartesian products are not Seifert manifolds. Namely, the following result holds. Proposition 4. Let the $3$-manifold $Q^3$ be a connected sum of lens spaces (the projective space $\mathbb{R}P^3$ is also treated as the lens $L(2,1)$) and Cartesian products $S^1\times S^2$ such that this connected sum contains at least two terms and, moreover, $Q^3$ is not homeomorphic to a connected sum to two copies of $\mathbb{R}P^3$. Then $Q^3$ is not a Seifert manifold. Thus the class of isoenergy surfaces of integrable billiard books is not covered by the class of Seifert manifolds. On the other hand this class contains examples of quite non-trivial Seifert manifolds. For example, consider the triangular billiard table bounded by an arc of an ellipse, and arc of a hyperbola, and the focal line. We glue two copies of this table along all components of the boundary. An isoenergy surface of the resulting billiard system is a spherical Seifert manifold with three singular fibres of type $(2,1)$, which correspond to the corners of the triangular billiard domain. 2.4. Billiards that are Liouville equivalent to classical integrable casesThe calculation of the Fomenko–Zieschang invariants of topological billiards and billiard books has allowed one to see that in many cases they coincide with the invariants calculated previously for integrable cases of rigid body dynamics (the Euler, Lagrange, Kovalevskaya, Joukowsky, Goryachev–Chaplygin–Sretenskii, Kovalevskaya–Yehia, Clebsch, and Sokolov systems). This allows one to prove that integrable cases in rigid body dynamics are Liouville equivalent to integrable billiard books. The papers [49] and [79] contain a list of known Liouville equivalent foliations and indicates regions in the bifurcation diagrams for the Euler, Lagrange, Kovalevskaya, Joukowsky, and Goryachev–Chaplygin–Sretenskii cases with such isoenergy $3$-surfaces. For each invariant a billiard system is indicated that models the behaviour of the closures of solutions on these isoenergy surfaces. Here we consider closely the cases of full realization (for the Euler and Lagrange systems) and partial realization (for the Joukowsky system) of integrable Hamiltonian systems in rigid body dynamics by means of integrable billiards. Recall that the system of motion of a rigid body hinged at a fixed point is defined on the dual space of the Lie algebra Ли $e(3)$ of the motion group of Euclidean space $\mathbb{R}^3$, that is, on the six-dimensional space $\mathbb{R}^6(S_1,S_2,S_3,R_1,R_2,R_3)$ with the following Poisson bracket (here $\varkappa=0$ and $\varepsilon_{ijk}$ is the sign of the permutation $(1,2,3) \to (i,j,k)$):
$$
\begin{equation*}
\{S_i, S_j\}=\varepsilon_{ijk} S_k, \qquad \{S_i,R_j\}=\varepsilon_{ijk} R_k, \qquad \{R_i, R_j\}=\varkappa\varepsilon_{ijk} S_k.
\end{equation*}
\notag
$$
Note that the case $\varkappa \ne 0$ corresponds to analogues of mechanical systems on other Lie algebras: on $\operatorname{so}(3,1)$ and on $\operatorname{so}(4)$. Such systems have also been closely investigated from the standpoint of Liouville foliations for them (see [88]–[91], [93], [114], and [115]). The two first integrals of this systems are Casimir functions, the geometric integral and the area integral We assume without loss of generality that $f_1=1$. Then the Liouville foliation on the symplectic sheet $M^4_g$: $f_1=1$, $f_2=g$ can depend on $g$ non-trivially.The Euler case (1750) describes the system of the dynamics of a rigid body hinged at its centre of mass. In this case the energy $H$ and the first integral $F$ have the following form (for principal moments of inertia $0< A_1 < A_2 < A_3$):
$$
\begin{equation*}
H=\frac{S_1^2}{2A_1}+\frac{S_2^2}{2A_2}+\frac{S_3^2}{2A_3}\,, \qquad F=F_{\rm e}=S_1^2+S_2^2+S_3^2.
\end{equation*}
\notag
$$
The formulae for the critical set of the system and the form of the bifurcation diagram of the map $(f_2,H)$ separating the ranges of values $(g,h)$ corresponding to different types of isoenergy manifolds $Q^3$ (see Figure 10), as well as the bifurcation diagram of the moment map $(H,F)$ in its dependence on the value of $g$ (the cases $g=0$ and $g \ne 0$) were presented, for instance, in Vol. 2 of [6]. The topological invariants of the system were also calculated there. For each of the resulting chambers shown in Fig. 10, an integrable billiard Liouville equivalent to the Euler system on the corresponding surfaces $Q^3_{gh}$ was found previously (see [49]). In Fig. 10 we present the required billiards and the Fomenko–Zieschang invariants describing the Liouville foliations for them. Here we have to indicate an ‘own’ billiard system for each isoenergy system. The Legendre case describes the motion of an axially symmetric rigid body with fixed point on the symmetry axis. The energy integral $H$ and the additional integral $F$ have the following form:
$$
\begin{equation*}
H=\frac{S_1^2}{2 A}+\frac{S_2^2}{2 A}+\frac{S_3^2}{2 B}+\varphi(R_3),\qquad F=F_{\rm L}=S_3.
\end{equation*}
\notag
$$
Depending on the potential $\varphi$ and the values of $f_2$ and $H$ there exist five types of isoenergy surfaces. To realize this system we can use circular topological billiards (when planar sheets are discs or annuli bounded by concentric circles; see Fig. 11): $S^3$: the table is glued of a disc and an annulus, along the outer circle of the annulus; $S^1 \times S^2$: the table is glued of two annuli, along their outer circles; $\mathbb{R}P^3$: the table is glued of two discs along their boundaries (in Fig. 11 one disc is partitioned into a smaller disc and an annulus); $S^3 \cup(S^1\times S^2)$: we use the table for $S^3$ described above and an annulus whose inner radius is not less that the outer radius of the table for $S^3$; $2 S^3$: we use the table for $S^3$ described above and a disc of radius not exceeding that of the inner circle of the first table. Now we go over to the Joukowsky case, which is a generalization of the Euler case reducing to the addition of a constant gyrostatic moment $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ (so that, in place of $S_i^2$, the Hamiltonian contains $(S_i-\lambda_i)^2$). The additional integral of the Joukowsky system is the same as for the Euler top: $F=F_{\rm e}$. The general form of bifurcation diagrams for the Joukowsky case is shown in Fig. 12, (a). In Fig. 12, (b) and (c), we present two special case of this bifurcation diagram. Dashed lines separate different Liouville foliations on homeomorphic isoenergy surfaces.  Figure 13.The Joukowsky system with bifurcation diagram of the type shown in Fig. 12, (b), for the map $(f_2,H)$ to the $Ogh$-plane. Regions in the $Ogh$-plane where the system can be modelled by one of the billiard systems $\alpha,\dots,\epsilon$. The vertical lines $A$, $B$ and $C$ code the same-name symplectic sheets. For systems of the type shown in Fig. 12, (b), in the $Ogh$-plane we distinguish the chambers where the Liouville foliation on the isoenergy surfaces $Q^3_{gh}$ is modelled by billiards (see Fig. 13). The construction of these billiards (which we denote by characters $\alpha,\dots,\epsilon$) is as follows:
Table 4.Cases of power lowering
Another vibrant topic is modelling mechanical systems (the Kovalevskaya top and Goryachev–Chaplygin systems) that possess polynomial integrals of high degree (3 or 4) with respect to momenta, by means of integrable billiards with the same canonical integral of degree $2$. Bolsinov and Fomenko showed [52] that for many systems (including the Kovalevskaya and Goryachev–Chaplygin systems) the degree of the first integral cannot be lower in the class of smooth Liouville equivalences. Theorem 15 (Fomenko and Vedyushkina [51]). The integrable systems of Goryachev–Chaplygin–Sretenskii, Kovalevskaya [6], and Kovalevskaya–Yehia [83], the Kovalevskaya system on the Lie algebra $\operatorname{so}(4)$ [89]–[91], the Sokolov [81] and Dullin–Matveev systems [116], with first integrals of degrees $3$ and $4$, can be realized in appropriate energy ranges (that is, are piecewise smooth Liouville equivalent) by integrable billiards with quadratic first integral. In other words, first integrals of high degree reduce, ‘in the piecewise smooth sense’, to the same quadratic integral Table 4 presents the correspondence between billiard tables, Fomenko–Zieschang invariants realized by means of these tables, integrable systems possessing integrals of high degree (the numbers of integrals and energy ranges are indicated in brackets in the notation from [6], [81], [83], [89]–[91], and [116]), and the topological types of the manifolds $Q^3$. 3. Evolutionary billiards and billiard equivalence of integrable systemsThe construction of an evolutionary (or force) billiard was proposed by Fomenko [57]. It enables one to model a system to both sides of a singular value of $h$ without modelling the foliation on the singular surface (which usually contains an equilibrium or a degenerate singularity). As the parameter of an evolutionary billiard (the velocity of the billiard ball or the ‘force’ of hits against the boundary edge) changes, both the topology of the billiard table and the law of ball reflection can change. For a singular value $h$ of the energy each foliation on $Q^3_{h\pm \varepsilon}$ is modelled by a separate billiard book, but the two books deform one into the other in accordance with certain rules. 3.1. The definition of an evolutionary billiardThe following definitions were introduced by Fomenko in [57] and [58] (also see [59]). Definition 16. 1. The support of a force billiard is a finite connected locally flat CW-complex $X$ (with Euclidean metric inside the $2$-cells). Its $2$-sheets $L_i$ are homeomorphic to closed simply connected domains in $\mathbb{R}^2$ and are bounded by a piecewise smooth curve with angles equal to $\pi/2$ at corners. $2$-sheets are glued by means of an isometry of their common smooth boundary arc (a spine of the book). 2. For each value of the energy parameter $H=h\geqslant0$ consider a closed (not necessarily connected) subcomplex $X(h)$ of the support $X$. We call it the state of the force billiard corresponding to the value $h$. Here $X(h_1)\subset X(h_2)$ for any $h_1 < h_2$ and $X=\bigcup X(h)$, where the union is taken over all values of $h$. Thus, the state $X(h)$ ‘grows’ with $h$. 3. We call (finitely many) values $h=1,\dots,N$ of $H$ at which the topology of the table or the law of reflection/refraction on boundary edges changes singular values, while other values are said to be regular. Recall that $1$-edges (spines) of $X(h)$ are arcs of confocal quadrics or concentric circles. 4. We denote the law or feflection/refraction on a spine edge $r$ in a state $X(h)$ by $Z(h,r)$. It is defined by a cyclic permutation on the $n$ sheets glued together along $r$, and it determines the dynamics of the particle after bouncing off the boundary. Let the system $Z(h)=\{Z(h,r)\}$ of this laws be a piecewise constant function of the energy which can make jumps only for singular values of $h$. 5. Assume that spine edges of $X(h)$ change smoothly in the class of confocal non-degenerate quadrics. It is known from the theory of integrable billiards that the resulting billiard systems are equivalent. For singular values of $h$, edges can glue with other edges, degenerate, or transform into intervals of the focal line. Sheets can glue along an arc of the same quadric. The new permutation cycle arises on the ‘new spine’. At a jump the billiard can change its equivalence class. For example, a piece of the boundary can lay down on the focal line for singular $h$, or it can fold in half. An angle of $\pi/2$ can become equal to $\pi$ in a jump. Spines of the same state $X(h)$ are allowed to glue together at an endpoint if they occur on the same arc, that the angle between them is $\pi$. In circular billiards boundary circles can contract to a point. For instance, in Fig. 14, at each of the two singular values of $h$ the corresponding edge $r$ becomes ‘permeable’ (‘transparent’). After that the billiard ball penetrates it, whereas before this time it bounced off. Then the pair of identity permutations on elliptic arcs is replaced by a transposition on an interval of the focal line. 6. Thus, the support $X$ is set to be constant, fixed. Within it the states $X(h)$ grow in size, and $X$ coincides with the terminal state $X(N)$. We call an integrable system with two degrees of freedom that describes the dynamics of a billiard ball on the variable states $X(h)$ a force (evolutionary) billiard. In paragraphs 7 and 8 below we assume that $h$ is a regular value of the energy in some interval $D_i=(i,i+1)$. The corresponding state (billiard table) is denoted by $X(D_i)$. 7. A point ni the phase complex $TX(D_i)$ is a pair $(x,v)$, where $x$ is a point on the billiard table $X(D_i)$ and $v$ is the velocity vector of the point mass at $x$. When $x$ occurs on the boundary of a sheet $L_i$ adjacent to another sheet $L_k$, the corresponding pairs $(x,v)$ and $(x,w)$ are glued in accordance with the reflection/refraction law $Z(h,r)$ that is in action on the edge $r$ for this value of $h$. 8. By a regular isoenergy $3$-surface $Q_h$ is the evolutionary billiard we mean the subset $H=h$ of the four-dimensional phase complex $TX(D_i)$. For book states such surfaces are topological $3$-manifolds. 3.2. Modelling by evolutionary billiards and the billiard equivalence of the Euler and Lagrange topsWe described the Euler and Lagrange tops above. The topology of a symplectic sheet $M^4_g$ for the Euler top is independent of the value of the area integral $f_2=g \ne 0$. In Fig. 14, (b), $M^4_g$ is the preimage of the vertical line under the map $(f_2,H)$. Singular isoenergy surfaces $Q$ are mapped to points on the three parabolas. A homeomorphism class of a regular surface $Q^3_{g,h}$ is assigned to each $2$-domain. Now we construct a force billiard for such $M^4_g$ (see Fig. 14). Its support is glued of two domains bounded by an ellipse and is homeomorphic to a $2$-ellipsoid $E^2$. States of this billiard are shown in Figure 14, (a), as subsets of $E^2$, and in Fig 14, (b), as gluings of plane domains. States with lower energy are positioned below and ones with higher energy are are above them. The initial (starting) state is a disconnected billiard table disjoint from the focal line (the lower picture). It is homeomorphic to two discs and realizes the Euler system on a pair of $3$-spheres $S^3$. Next it transforms into an annulus, realizing the product of a $2$-sphere and a circle $S^1\times S^2$. Then the annulus turns to a sphere (an ellipsoid) and realizes the projective space $\mathbb{R}P^3$. In Fig. 14, (b), we also show trajectories of the billiard ball and gluings of spines. That at each moment of its evolution the billiard is integrable because its boundaries lie on arcs of confocal quadrics. Theorem 16 ([57], [58]). An integrable force billiard whose support is homeomorphic to an ellipsoid realizes (in the sense of Liouville equivalence) the integrable Euler case on the whole of the phase manifold $M^4_g$ at the same time, apart from the singular levels of energy, that is, on all regular isoenergy $3$-surfaces for all regular values of $g$ and $h$. Systems of the Lagrange top can have — depending on the value of the area integral, the relation between the moments of inertia, and the choice of the potential — four types of bifurcation diagram (see [6]) and precisely five types of symplectic $4$-sheets. For all of them we found force billiards [58]. We show one of these in Fig. 15. Theorem 17 ([57], [58]). On each regular symplectic $4$-sheet $M^4_g$ the Lagrange integrable case can be realized (in the sense of Liouville equivalence) by one of the five force billiards constructed in [57]. Their force billiards are bounded by concentric circles (and therefore integrable at each moment of their evolution). On the level of evolutionary billiards modelling the Euler and Lagrange tops we were successful in discovering a non-trivial link between these systems. Let us deform systems of confocal ellipses and hyperbolae into a family of concentri circles and radial rays (by letting the foci tend to each other). Theorem 18 ([57], [58]). The above deformation of confocal billiards into circular ones transforms the force billiard realizing the Euler case into another force billiard, the full system of Liouville foliations for which coincides with the full system of foliations for the Lagrange case (obtained by combining all three types of isoenergy $3$-surfaces). We call systems described in Theorem 18 billiard equivalent systems. The ‘transformation’ of the Euler case into the Lagrange case, which we discovered, does not transform a symplectic sheet of the Euler case into one of the five types of symplectic sheets of the Lagrange case. This is more complicated, which was previously an obstruction to detecting a transformation of these systems one into the other. Nevertheless it turns out that the full system of Liouville foliations for the Euler case transforms into the full system of Liouville foliations for the Lagrange case. This required us, first of all, to discover ‘hidden confocal quadrics’ in the Euler case and ‘hidden concentric circles’ in the Lagrange case. As a result, a deformation of confocal quadrics into circles (as the foci merge) is what transforms the Euler case into the Lagrange one. 3.3. Partial modelling for the Kovalevskaya and Joukowsky systemsConsider the classical Kovalevskaya top. Its phase topology was considered by Kharlamov in [80], where he constructed bifurcation diagrams and determined the homeomorphism class of leaves of the Liouville foliation. The rough molecules for this top were calculated by Oshemkov [117], and the marks on the Fomenko–Zieschang invariant were calculated in [56]. The bifurcation diagram of the map $(f_2,H)$ is shown in Fig. 16: continuous lines separate domains $Q^3$ with different topology, and dashed lines correspond to the presence of a non-Bott singularity of rank $1$ in the corresponding $Q^3$, that is, they separate domains with the same homeomorphism classes but different Liouville foliations (described by Fomenko and Fomenko–Zieschang invariants). Consider the symplectic sheet $A$ corresponding to the vertical line $g=\operatorname{const}$ shown in Fig. 16. This line intersects five chambers of the bifurcation diagram, and the foliations on isoenergy surfaces for the first three chambers can be modelled by foliations for integrable billiards bounded by arcs of confocal quadrics. These billiards, denoted by $\alpha$, $\beta$, and $\gamma$, are shown in Figure 16. They are constructed as follows. $\alpha$. An elementary billiard domain $\alpha$ is bounded by an arc of an ellipse (we set the corresponding value of $\lambda$ to be $0$), a non-convex arc of a hyperbola (we set the corresponding value of $\lambda$ to be $b+(a-b)/2=(a+b)/2<a$), the degenerate hyperbola $x=0$ (recall that the corresponding value of $\lambda$ is $a$) and the focal line (recall that the corresponding value of $\lambda$ is $b$). $\beta$. A topological billiard $\beta$ is glued of two billiards $\alpha$, where the gluing occurs along both arcs of hyperbolae. $\gamma$. A billiard book $\gamma$ is glued of six elementary billiard domains. Billiards with indices $1$ and $2$ are bounded by arcs of the ellipse with parameter $\lambda=0$ and hyperbola with parameter $\lambda=(a+b)/2$. Billiards with indices $3$ and $4$ are bounded by arcs of the ellipse with parameter $\lambda=0$ and the hyperbola with parameter $\lambda=(a+b)/2$, and by segments of the degenerate hyperbola with parameter $\lambda=a$. Billiards with indices $5$ and $6$ are bounded by an arc of the ellipse with $\lambda=0$ and an arc of the hyperbola with $\lambda=a$. To the arcs of hyperbolae which are spines of the book we assign the following perturbations: the permutation $(1,3,2,4)$ to the arc of the hyperbola with parameter $\lambda=(a+b)/2$ and the permutation $(3,5,4,6)$ to arc of the degenerate hyperbola. Note that the billiard domain $\alpha$ constructed above is a subset of $\beta$, while $\beta$ is in its turn a subset of $\gamma$. This allows one to regard them as states of a single force billiard $K$. As an ambient billiard complex we take the billiard book $\gamma$ and as the initial state we take $\alpha$, whose image under the isometric embedding in $\gamma$ covers the upper half of the sheet with index $3$. We embed the intermediate state, the billiard domain $\beta$, in the book $\gamma$ so that its image under this isometric embedding covers the upper halves of $3$ and $4$. Now we describe bifurcations of the force billiard $K$ we have constructed. At the first jump, when $\alpha$ is transformed into $\beta$, a billiard sheet is added and hyperbolic edges become partly permeable. Now the billiard ball moves along sheets $3$ and $4$ of the billiard book $\gamma$, but it still cannot penetrate the focal line. At the second jump billiard sheets with indices $1$, $2$, $5$, and $6$ are added, the permutations assigned to hyperbolic arcs change, and the focal line becomes permeable. Theorem 19 ([59]). The evolutionary billiard $K$ constructed above realiz s (in the sense of Liouville equivalence) the Kovalevskaya integrable case on a part of the symplectic phase manifold $M^4_g$ corresponding to the line $A$ in Fig. 16. We stress that edges of the billiard domain evolve in the class of confocal quadrics, which ensures that the system is integrable at each moment of its evolution, on all isoenergy $3$-surfaces arising sequentially with the growth of energy. Now we go over to the problem of modelling the Joukowsky system by an evolutionary billiard. We construct evolutionary billiards $J_A$, $J_B$, and $J_C$ which model it in part on the symplectic sheets $A$, $B$, and $C$. The billiard $J_A$. The line $A$ intersects successively the chambers $\alpha$, $\beta$, $\gamma$, and $\delta$ in the $Ogh$-plane. The initial state of the evolutionary billiard $J_A$ is the billiard domain $\alpha$ and the terminal state is $\delta$. At the first jump of energy, to the billiard $\alpha$ a copy of it is added, which is glued to it at the next two jumps. First the gluing goes along the focal line and then along the arc of the smaller ellipse, the one with parameter $\lambda=b/2$. The billiard $J_B$. The symplectic sheet $B$ goes successively through the chambers $\alpha$, $\gamma$, and $\delta$ in the $Ogh$-plane. The initial state of the force billiard $J_B$ is the billiard domain $\alpha$, an intermediate state is $\gamma$, and the terminal state is $\delta$. By contrast to the previous force billiard, at the first jump both a copy of $\alpha$ arises and the two copies are glued along the focal line. The billiard $J_C$. The symplectic sheet $C$ goes successively through the chambers $\alpha$, $\epsilon$, and $\delta$ in the $Ogh$-plane. In traversing the chamber $\alpha$ the same-name billiard table varies so that the the parameter $\lambda$, equal to $b/2$ originally, tends to $b$. In the limit, when the boundary of the chamber is attained, the state of the billiard is two copies of the ‘bottom’ sheet $\alpha$. At the moment of jump the two copies are glued together along the focal line. At the next jump two billiard sheets, glued along an arc of the ellipse with parameter $b/2$, are added, which are attached to the previous state of the billiard along the arc of the larger ellipse. Then one convex gluing edge is replaced by two convex and one non-convex one. On the other hand this jump can be interpreted as ‘squeezing a fold’ inside the billiard out of two new small billiard domains (see the shapes of the billiards $\epsilon$ and $\delta$ in Fig. 13), when in the whole small neighbourhood of the jump the billiard remains homeomorphic to an annulus, and only the number of gluings grows. In [59] these authors also performed partial modelling of the Kovalevskaya and Joukowsky systems, for which the arcs and vertices of bifurcation diagrams have positions different from what we discussed above. 3.4. Billiard equivalence of geodesic flowsBy means of topological billiards and billiard books we can model geodesic flows on two-dimensional surfaces. By Kozlov’s theorem [9], [54], if a geodesic flow on a closed two-dimensional surface is integrable, then the Euler number of this surface is non-negative. Under the assumption of orientability, a surface satisfying the hypotheses of the theorem can either be a torus $T^2$ or a sphere $S^2$, and in the non-orientable case it can be a Klein bottle $\operatorname{KL}^2$ or a projective plane $\mathbb{R}P^2$. To date several authors (Kolokoltsov [118], Matveev [119], and Babenko and Nekhoroshev [120]) obtained a classification of all integrable flows on these four surfaces such that the additional first integral is polynomial in momenta and has degree at most two. In other words, they described all linearly and quadratically integrable geodesic flows on closed two-dimensional surfaces. For all of these flows the canonical form of the metric was indicated and the Fomenko–Zieschang invariants were calculated (see Matveev [119], Selivanova [121], Kalashnikov (jr.) [122], Zung, Polyakova, and Selivanova [123]). Recently Vedyushkina and Fomenko showed [55] that, given a closed orientable surface and an arbitrary geodesic flow on it which is linearly or quadratically integrable, we can algorithmically construct a billiard (a topological billiard or a billiard book) such that the geodesic flow and the billiard have the same Fomenko–Zieschang invariants. Theorem 20 (Vedyushkina and Fomenko [55]). 1) Each geodesic flow on an orientable two-dimensional surface (a torus or a sphere) that admits a linear first integral is Liouville equivalent to a topological billiard formed by planar billiard domains bounded by concentric circles. 2) Each geodesic flow on an orientable two-dimensional surface (a torus or a sphere) that admits a quadratic first integral is Liouville equivalent to a topological billiard or a billiard book formed by planar billiards bounded by confocal ellipses and hyperbolae. 3) These topological billiards can explicitly and algorithmically be constructed on the basis of the parameters of the integrable metric. Billiards modelling linearly integrable flows can be obtained as follows. We call billiard ring domains billiards of type $C$ and billiard disc domains billiards of type $D$. All these billiards are bounded by circles from the same family of concentric circles. Consider a gluing of such billiard domains such that the resulting topological billiard is homeomorphic to a sphere $S$ (in which case it includes two discs $D$) or a torus $T$ (in which case it consists only of annuli $C$). We present examples of such sets of billiard domains in Fig. 17.  Figure 17.Examples of billiards modelling linearly integrable geodesic flow on a sphere and a torus. Given a topological billiard, we define the following piecewise linear function. We number all planar billiards forming it. In the case of a billiard $S$ homeomorphic to a sphere, first we take one of the two elementary billiards homeomorphic to a disc. On the $Oxy$-plane we distinguish the points $(0,r_0)$ and $(0,r_1)$, where $r_0$ and $r_1$ are the smaller and greater radii of the circles bounding the first billiard domain (if its has type $D$, then we set $r_0=0$). Going over to the next billiard, we distinguish the points of the form $(i,r)$, where $i$ is the index of the billiard domain and $r$ is the radius of the boundary circle along which it is glued to the billiard domain next in order. We join all these points by a polygonal curve. Let $\widetilde{f}$ denote the piecewise linear function whose graph coincides with this curve. Note that the graph of $\widetilde{f}$ is the ‘profile’ of the billiard. Next we foliate the domain under the graph of $\widetilde{f}$ by horizontal line segments. Contracting each segment to a point we obtain a graph. To vertices of this graph corresponding to local maxima of $\widetilde{f}$, that is, to convex gluing edges, we assign atoms $A$. To vertices of the graph corresponding to local minima of $\widetilde{f}$ that is, to non-convex gluing edges of billiard domains, we assign atoms in the series $B_k$, where $k$ is the positive integer which is one less than the number of billiards participating in the gluing (that is, the number of gluing edges on the circle of the relevant radius). If the billiard is homeomorphic to a torus, then to the global minimum of $\widetilde{f}$ we assign an atom from the series $C_k$. For a billiard $S$ the resulting graph has one free edge, that is, an edge without an atom on its endpoint vertex; we denote this graph by $W(\widetilde{f})$. For a torus $T$ the resulting graph has two free edges; we denote it by $W_2(\widetilde{f})$. Fix an arbitrary triple $(q,t,L)$, where $t\in[0,1)$, $L>0$, and $q(v)$ is a function with period $L$. For this triple we can construct a Riemannian metric on the torus. To do this, in the plane with Cartesian variables $(u,v)$ we consider the metric and then take the quotient of $\mathbb{R}^2$ by the lattice generated by the vectors $f_1=(1,0)$ and $f_2=(t,L)$. We call such metrics $(q,t,L)$-metrics. They admit linear first integrals. Furthemore, the following result holds.Proposition 5 (Matveev [119]). Assume that the geodesic flow of a metric on a torus admits a linear integral. Then this metric is either flat or isometric to a $(q,t,L)$-metric. Tow metrics corresponding to triples $(q,t,L)$ and $(\widehat{q},\widehat{t},\widehat{L})$ are isometric if and only if their parameters satisfy the relations indicated in [119]. Theorem 21 (Selivanova [121]). Assume that the geodesic flow of a metric $ds^2$ on a torus $T^2$ is linearly integrable (so that $ds^2$ is a $(g,t,L)$-metric). Let $W_2(g)$ be the graph constructed for the function $g$. Then the marked molecule $W^*$ corresponding to the geodesic flow of this metric has the form shown in Fig. 18, (a). Marks are put as follows: all edges not containing atoms $A$ carry the mark $r=\infty$; edges containing an atom $A$ carry the mark $r=0$; the only family present carries the mark $n=0$; the marks $\varepsilon$ on the edges $a$ and $b$ are equal to $-1$, while on all other edges they are equal to $+1$ (see Fig. 18, (b)). The idea of the proof of Theorem 20 for integrable geodesic flows with linear first integral on a torus consists in constructing, from the positive function $g$ in the definition of the $(g,t,L)$-metric on the torus, a piecewise linear function $\widetilde{g}$ such that the mutual position of its maxima and minima coincides with the mutual position of the maxima and minima of $g$, which implies that the graphs $W_2$ for these functions are the same. The marks also coincide, as follows from results due to Selivanova [121] and Vedyushkina, who calculated the corresponding Fomenko–Zieschang invariants. Definition 17. A Riemannian metric on a torus is said to be globally Liouvillean if there exist global covering coordinates $x$ and $y$ on the torus in which this metric has the form where $f(x)$ and $g(y)$ are non-constant smooth positive functions with periods $T_x$ and $T_y$, respectively. Theorem 22 (Selivanova [121]). Let $ds^2$ be a globally Liouvillean metric on the torus $T^2$ defined by two functions $f$ and $g$. Let $W_2(f)$ and $W_2(g)$ be the graph constructed for $f$ and $g$. Then the marked molecule $W^*$ corresponding to the geodesic flow of this metric has the form shown in Fig. 18, (b). The marks are defined as follows. The four edges $a$, $b$, $c$, and $d$, as well as the edges containing an atom $A$ carry the mark $r=0$. All other edges have the mark $r=\infty$. All marks $n$ are equal to zero and all marks $\varepsilon$ are equal to $1$. Assume that a geodesic flow on a torus has a first integral that is quadratic in momenta and the metric on the torus is globally Liouvillean. Using the result of Fomenko and Vedyushkina in [55] we construct algorithmically a confocal billiard book (where the planar sheets of the CW-complex are bounded by arcs of confocal quadrics) such that the geodesic flow and the billiard system have the same Fomenko–Zieschang invariants. Next we let the foci of this billiard book tend to each other. Then the planar sheets of the confocal billiard book (bounded by arcs of confocal ellipses and hyperbolae) turn to planar sheets of the limiting billiard book, which are bounded by arcs of concentric circles and radial lines. The centre of these circles is at the origin, which is the limit of the pair of foci of ellipses and hyperbolae. The Fomenko–Zieschang invariant of the billiard system on the limiting circular billiard turns out to be the Fomenko–Zieschang invariant for the system of an integrable geodesic flow on a 2-torus which has a linear first integral, while the canonical form of the metric is determined by the one for the original geodesic flow with a quadratic first integral. More precise;y, the following result holds. Theorem 23. Given the geodesic flow of a Liouvillean metric $ds^2=(f(x)+g(y))(dx^2+dy^2)$ on a $2$-torus, consider a billiard bounded by arcs of confocal quadrics whose Fomenko–Zieschang invariant coincides with that of the geodesic flow, and let a level set $\Lambda<b$ for this billiard correspond to the subgraph $W_2(f)$ of the invariant. Then as the foci of the billiard approach each other, the billiard transforms into another billiard which models the integrable geodesic flow on the torus with metric of the form $ds^2=f(x)(dx^2+dy^2)$. In this sense linearly integrable geodesic flows are billiard equivalent to quadratically integrable geodesic flows on the torus which are defined by a globally Liouvillean metric. Proof. To prove this theorem we explain how we can construct the required billiard from the functions $f$ and $g$.
For the construction of the invariant of a geodesic flow, rather then the functions $f$ and $g$ themselves, we need to know the mutual position of maxima and minima on their periods $T_x$ and $T_y$. The domain under the graph of the function $f$ is fibred by segments of horizontal lines and then each segment is contracted to a point. To terminal vertices of the resulting graph we assign atoms $A$, to the least local minimum an atom in the series $C$, and to all other minima atoms in the series $B$. Then we obtain the graph $W_2(f)$. Next we shrink the graph of $f$ so that the values of the function lie in the interval $(0,b)$ and consider the piecewise linear function $\widetilde{f}$ coinciding with the modified function $f$ at its extremum points. We do the same with the function $g$ and construct a function $\widetilde{g}$. Note that the graph $W_2(\widetilde{f})$ (the graph $W_2(\widetilde{g})$) is different from $W_2(f)$ (from $W_2(g)$, respectively) only by edge lengths. Now we construct a billiard domain homeomorphic to a torus and glued of billiards $B_0$. Recall that each elementary billiard domain $B_0$ is a quadrilateral bounded by two arcs of ellipses and two arcs of hyperbolae. We number all extremum points of $\widetilde{f}$ and denote them by $f_i$, $i\in\{1,\dots,m+1\}$. In a similar way we denote all extremum points of $\widetilde{g}$ by $g_j$, $j\in\{1,\dots,n+1\}$. Consider a set of domains $B_0$ each of which is bounded by two arcs of the ellipses with parameters $b-f_i$ and $b-f_{i+1}$ and two arcs of the hyperbolae with parameters $g_j$ and $g_{j+1}$. We glue this set of $mn$ billiards into a torus $T(B_0)$ in the order of the respective edges of the piecewise linear functions $\widetilde{f}$ and $\widetilde{g}$. The Fomenko–Zieschang invariant of the resulting graph contains the subgraphs $W_2(\widetilde{f})$ and $W_2(\widetilde{g})$ and coincides with the invariant of the geodesic flow (see more details in [55]). Now we let the foci of the confocal quadric approach each other. In the limit the billiard domain $T(B_0)$ turns to $T(C)$. Each billiard $B_0$ is transformed into a domain bounded by two circles and two radial lines. Note the following transformation of trajectories. If a trajectory (or its extension) was previously tangent to an ellipse, then this trajectory is now tangent to a circle. Note that, as before, all such trajectories fall into two classes, depending on the (clockwise or anticlockwise) direction in which they go around the origin. Trajectories on lines through foci turn to trajectories lying on radial lines. The billiard system $T(C)$ thus obtained has no other trajectories. Hence its rough Fomenko molecule consists of two copies of the graph $W_2(f)$ glued one to the other. Calculating the marks (quite similarly to calculating the marks for $T$) we see that the Fomenko–Zieschang invariant of $T(C)$ coincides with the Fomenko–Zieschang invariant for the geodesic flow on the torus with metric $ds^2=dx^2+f(x)dy^2$ (which defines a linearly integrable geodesic flow on a 2-torus). $\Box$ We turn to the case of a sphere. Let the torus $T^2$ be defined as the quotient of $\mathbb{R}^2$ with Cartesian variables $(x,y)$ by the orthogonal lattice $\Gamma$ with basis formed by the vectors $f_1=(1,0)$ and $f_2=(0,L)$, where $L$ is an arbitrary positive number. Consider the involution $\sigma$ of the torus induced the symmetry $\sigma(x,y)=(-x,-y)$ (symmetry with respect to the origin) of the covering plane. Clearly, $\Gamma$ is stable under this symmetry, so $\sigma$ is indeed an involution of the torus. Consider the natural projection $\xi\colon T^2\to T^2/\sigma$. Proposition 6. The quotient $T^2/\sigma $ of the torus by the involution $\sigma$ is homeomorphic to the $2$-sphere $S^2$. The projection is a branched covering of multiplicity two of a sphere with four branch points, each of which has a unique preimage on the torus. On the covering plane of the torus we define two periodic smooth functions $f(x)$ and $g(y)$ satisfying the following conditions. (a) The function $f(x)$ is non-negative, even, and smooth and has period $1$. (b) The function $g(y)$ is non-negative, even, and smooth and has period $L$. (c) (The asymptotic behaviour of $f(x)$ and $g(y)$ in a neighbourhood of their zeros.) The function $f(x)$ vanishes at the points of the form $x=m/2$, $k\in \mathbb{Z}$. The function $g(y)$ vanishes at the points of the form $y=kL/2$, $k\in \mathbb{Z}$. For each point of the form $(m/2,kL/2)$ there exists a smooth function $h(t)$ in a neighbourhood of zero such that $h(0)=0$, $h'(0)\ne 0$,
$$
\begin{equation*}
f\biggl(\frac{m}{2}+t\biggr)=h(t^2)\quad\text{and} \quad f\biggl(\frac{kL}{2}+t\biggr)=-h(-t^2).
\end{equation*}
\notag
$$
The next proposition describes all quadratically integrable geodesic flow on a 2-sphere. Proposition 7 (Zung, Polyakova, and Selivanova [123]). 1) Let $(f(x)+g(y))(dx^2+dy^2)$ be a metric on the torus $T^2$ satisfying conditions (a)–(c), and let $\xi\colon T^2\to T^2/\sigma$ be the two-sheeted covering described above. Then there exists a unique smooth Riemannian metric $ds^2$ on the sphere $S^2$ such that If, in addition, $f$ and $g$ are real analytic, then the metric $ds^2$ is too. 2) Conversely, consider the metric $\xi^*(ds^2)$ on $T^2$, where $ds^2$ is a smooth metric on $S^2$. If it has the form then the functions $f$ and $g$ satisfy conditions (a)–(c).A metric on the sphere has a linearly integrable geodesic flow if and only if there exist global conformal coordinates $x$ and $y$ in which the covering metric on the torus has the form where $f(t)$ is a smooth positive function on the half-axis $[0,+\infty)$ such that $f(1/t)/t^2$ is a positive smooth function on the whole of $[0,+\infty)$ (that is, also at zero).This can conveniently be formulated as follows. Proposition 8. A metric $ds^2$ on the sphere has a linearly integrable geodesic flow if and only if there exist smooth global coordinates $(\theta,\varphi)$ with two singular points (analogues of poles for ordinary spherical coordinates) on the sphere, where $\theta$ ranges between $0$ and some $\theta_0$, and $\varphi$ is a periodic coordinate, $(0\leqslant \varphi\leqslant 2\pi)$, such that in these variable the metric has the form Note that the Hamiltonian $H$ of the geodesic flow has the form and the first integral is $F=p_\varphi$.Theorem 24. Consider a geodesic flow on a $2$-sphere defined by a $(L,f,g)$-metric such that the function $g$ has a unique maximum on the period. Consider a billiard modelling this geodesic flow and bounded by arcs of confocal ellipses and hyperbolae. Let the foci of this billiard approach one the other. Then the limiting billiard models a linearly integrable flow on the $2$-sphere defined by the metric $ds^2=d\theta^2+f(\theta)\,d\varphi^2$. Thus, the class of linearly integrable geodesic flows on a $2$-sphere is billiard equivalent to a subclass of square integrable geodesic flows on the $2$-sphere. Proof. Consider a billiard modelling a square-integrable geodesic flow on a $2$-sphere and defined by the $(L,f,g)$-metric from the hypothesis. In general position such a billiard (see Fig. 19) contains non-convex gluing edges lying on ellipses whose position is determined by the function $\widetilde{f}$ (which is constructed from $f$ similarly to the case of a $2$-torus) and non-convex gluing edges lying on hyperbolae whose position is determined by the function $\widetilde{g}$ (constructed from $g$). Since $g$ has no minimum points on the period, the billiard domain modelling this geodesic flow and constructed in accordance with the algorithm from the proof of the theorem on modelling geodesic flow (Theorem 20) is glued of elementary billiards, so that it has no non-convex gluing edges lying on hyperbolae. Then the marked molecule coding the Liouville foliation for this billiard has the form indicated in Fig. 20, (a) (provided that the billiard domain contains no non-convex gluing edges at all) or in Fig. 20, (b) (if it contains non-convex gluing edges lying on ellipses).
Now we let the foci tend to one point (the origin). Then confocal ellipses become concentric circles and hyperbolae turn to straight lines through the origin. Furthermore, the billiard system continues to be integrable, and the molecule coding the Liouville foliation on the corresponding isoenergy surface becomes either $A$–$A$ (with marks $r=0$ and $\varepsilon=1$) or the molecule shown in Fig. 20, (c). All such molecules are the full sets of invariants coding linearly integrable geodesic flows on a 2-sphere. Remark 8. Assume that a geodesic flow on a 2-sphere is defined by arbitrary functions $f$ and $g$ such that $g$ has minimum points on the period. Then the billiard modelling this geodesic flow contains non-convex gluing edges lying on some hyperbolae. Hence, as the foci approach one the other, a neighbourhood of the origin is glued in the limit of sectors formed by the asymptotes to these hyperbolae. In such a billiard one cannot define the motion of a point mass occurring at the origin by continuity. Thus, such geodesic flows do not turn to linearly integrable geodesic flows under the above transformation. 4. Integrable generalizations of planar billiards and billiard booksIn this section we discuss several generalizations of confocal and circular billiards in which systems are still integrable. First, this is the introduction of a polynomial potential (in particular, a Hooke potential for a confocal or a circular billiard), the introduction of a permanent and constant in space magnetic field on a circular billiard table, whose induction vector is orthogonal to the table, and the addition of slipping of the particle along the boundary of the table (billiards with slipping). It is easy lt verify that gluing a billiard book of planar tables with such dynamics preserves the integrability of the system, so that the class of billiard books introduced by Vedyushkina can be combined with such generalizations. The new classes of billiards have successfully been used in modelling of complex integrable systems. For example, using planar and topological billiards with slipping Zav’yalov and these authors [63], [124] succeeded in modelling geodesic flows on the non-orientable surfaces $\mathbb{R}P^2$ and $\operatorname{KL}^2$. In a magnetic billiard systems it is possible to realize both the molecule $A$–$A$ with marks $r=\infty$ and $\varepsilon=-1$ and singularities of complexity $1$ splittable in the sense of Zung (see Vedyushkina and Pustovoitov [125], and also [47] and [126]). Billiard books with potentials were used for modelling arbitrary non-degenerate singularities of rank $0$. The question, raised by Fomenko and Vedyushkina [35], on modelling non- degenerate singularities of rank 0 (that is, of foliations in a neighbourhood of an equilibrium of the system or a level containing it) by means of billiards has also been answered in the affirmative. For singularities of centre-centre, centre-saddle, and saddle-saddle type the approach proposed and developed by Kibkalo [65], [66] uses confocal billiards (as limits of geodesic flows on an ellipsoid) and sets of permutations found by Oshemkov for saddle singularities of smooth systems. Focus–focus singularities were modelled by Vedyushkina, Kibkalo, and Pustovoitov [67] by means of billiard books with attracting Hooke potentials which are glued of $n$ discs with permutation $(1,\dots,n)$. We discss separately the question of the topology of the Liouville foliation for billiards in multidimensional domains, that is, domains of dimension 3 or higher. The case of a billiard domain bounded by an ellipsoid in $\mathbb{R}^3$ was considered by Dragović [100]. The classification of confocal domains on a three-dimensional ellipsoid up to rough equivalence was completed by Belozerov [68]. He also considered the topology of Liouville foliations for billiard systems in such domains in the case when a Hooke potential is added (papers in print). We also note briefly an integrable generalization of billiards and geodesic flows on quadrics descovered quire recently. Kibkalo showed that the geodesic flow on the intersection of $n-2$ confocal quadrics of different types in $\mathbb{R}^n$ is integrable. Also, a quadratic first integral independent of energy was indicated in the elliptic coordinates. Note that the metric on the metric on this two-dimensional surface can be reduced to a Liouvillean form. After that Belozerov proved a general result, which in fact generalizes the celebrated classical Jacobi–Chasles theorem: the geodesic flow on the intersection of $k$ confocal quadrics of different types in $\mathbb{R}^n$ is completely integrable, and tangent lines to each geodesic curve on this $(n-k)$-dimensional intersection are simultaneously tangent to $n-k-1$ quadrics, which are the same for all points in this intersection. The topology (homeomorphism class) of this intersection was also described. The proof is performed in the elliptic coordinates and uses some non-trivial approaches to symmetric polynomials. We mention briefly several new directions connected with the topology of integrable billiards. Dragović and Radnocić [127] and Karginova [128], [129] investigated integrable billiards in the plane with Minkowski metric that are also bounded by arcs of confocal quadrics. One can also add a central Hooke-type potential to this system preserving its integrability. One example was analysed by Skvortsov and Vedyushkina [130]. Assuming that the boundary of the billiard table can contain angles of $3\pi/2$ (with respect to the interior of the domain) we obtain a so-called pseudointegrable billiard. Rather than to a torus (as in the integrable case), a connected component of a regular common level set of first integrals is now homeomorphic in the general case to some surface with handles and several punctures. Here we can name papers by Dragović and Radnović [131]–[133] and Moskvin [134], [135], who considered such billiards. Note that the existence of a conditional integral (a measurable essentially non- constant function that keeps its value on almost all trajectories) for a billiard on a compact two-dimensional Riemannian manifold with elastic reflection at the boundary requires, according to Bolotin [136], then a certain non-strict inequality connecting the sum of angles at corners of the boundary and the Euler number of the surface holds. If this inequality turns to equality then all angles must be commensurable with $\pi$, with denominator equal to the degree in momenta of the first integral. It is also an interesting topic to model an arbitrary ordered billiard game [100] in the plane as the projection of trajectories of a ball traversing an algorithmically constructed billiard book: see [137]. For some of such book the Fomenko invariants (markless molecules) were also calculated in that paper. 4.1. Billiards with slippingThe reflection/refraction law at a point on the boundary (or on a gluing curve) considered above is the standard Huygence law of elastic reflection: the angle of incidence is equal to the angle of reflection and the velocity vector preserves its length. Another class of billiards, called billiards with slipping, was proposed by Fomenko in [63]. The idea of the construction is briefly as follows. After hitting the boundary at a point $x$ the point mass continued its motion from a point $y$ lying at some distance from $x$ along the boundary $\partial\Omega$. In this case the angle of reflection is equal to the angle of incidence in the following sense: the oriented angles between the velocity vector on the curve (the direction of the curve determines the direction of slipping) and the corresponding direction (of incidence or reflection) are measured: see the example shown in Fig. 21. Consider a centrally symmetric domain and slipping ‘through an angle of $\pi$’, when the trajectory extends from the diametrically opposite boundary point. For centrally symmetric confocal and circular tables such slipping gives rise to an integrable billiard. As itself, the introduction of slipping in an integrable case is equivalent to the introduction of non-orientable billiards. For instance, in the simplest case the introduction of slipping on the boundary of a disc (see Fig. 21, (a)) allows us to say that, in fact, we consider a billiard system on the projective plane which is obtained by identifying the opposite boundary points of the disc. Furthermore, it turns out that billiards with slipping model linearly integrable geodesic flows on the projective plane $\mathbb{R}P^2$ and Klein bottle $\operatorname{KL}^2$. Integrable geodesic flows on non-orientable surfaces are regarded as certain quotients of geodesic flows on orientable surfaces. Using this idea one can construct the required billiards algorithmically, from billiards modelling geodesic flows on a torus and a sphere. To do this, from such topological billiard domains one removes some subdomains and defines slipping on the boundary of what remains. The resulting billiard table is homeomorphic to a projective plane or a Klein bottle. Theorem 25 (Vedyushkina and Zav’yalov [124]). A geodesic flow on a non- orientable two-dimensional manifold (a Klein bottle or a projective plane) that has an additional first integral which is linear in momenta is Liouville equivalent to a billiard with slipping formed by planar billiards bounded by concentric circles. The linear first integral of this flow reduces to the canonical first integral of the billiard system, namely, the angle between the trajectory and the boundary of any of these billiard tables. For square integrable geodesic flows several examples have currently been examined. Here is one of these. Consider a billiard $Z$ glued of two annuli bounded by two confocal ellipses each so that slipping is introduced on each boundary component. Theorem 26 (Zav’yalov [63]). The Fomenko–Zieschang invariant of the topological billiard $Z$ is as shown in Fig. 21, (b). This billiard system is piecewise smoothly Liouville equivalent to an integrable geodesic flow on the Klein bottle $\operatorname{KL}^2$ which has a quadratic (in components of the momentum) additional integral. Slipping can also be introduced on billiard books. We construct a billiard book with slipping as follows. Consider $m$ billiard domains bounded by an ellipse and attach to them an annulus bounded by the same ellipse and another ellipse with a smaller value of the parameter $\lambda$ (that is, a larger ellipse). We number sheets of the book as follows: discs obtain indices $1,\dots,m$ and the annulus is assigned the index $m+1$. On the gluing curve (the boundary ellipse of the tables $A_2$ and the inner boundary ellipse of the annulus) we specify a cyclic permutation of length $m+1$ of all $2$-cells of the book. We add slipping to this book by defining it on the larger ellipse of the annuku (so that, in fact, we attach a Möbius band to the original complex): after hitting the larger ellipse the particle continues its motion from the point opposite to the hitting point, so that it ‘slips’ by an angle of $\pi$ along the boundary. The billiard book thus defined is integrable because links of trajectories are still tangent to some ellipse or some hyperbola. Let $\mathbb{B}_s(mA_2+C_2)$ denote this billiard table. In Fig. 22 we show a billiard trajectory schematically. Note that there is no reflection when the particle goes from sheet no. $m$ to annulus no. $m+1$: the point mass goes through the spine of the book. Proposition 9 (Vedyushkina). The Fomenko–Zieschang invariant describing the Liouville foliation of an isoenergy surface of the billiard book with slipping $\mathbb{B}_s(mA_2+C_2)$ is shown in Fig. 23, (a) for an even $m$, and in Fig. 23, (b) for an odd $m$. 4.2. Integrable billiards with potentialAnother integrable generalization of the classical billiard system in $\Omega \subset \mathbb{R}^2(x,y)$ is obtained by introducing a suitabe potential. A simplest example id the Hooke potential $\pm k(x^2+y^2)$ for a confocal or circular billiard table. One can also introduce such a potential on a billiard book glued of confocal (or circular) billiard domains [66]: the law of reflection from the boundary preserves the values of the additional integral and energy, the boundaries of domains of possible motion (the projections of Liouville leaves onto the table) lie on confocal quadrics (concentric circles), and the projection of the singular circle of each $3$-atom for such a system also lies on some quadric. The situation is similar for an entire class of polynomial potentials (Pustovoitov). Consider a billiard inside an ellipse $x^2/a+y^2/b=1$ with perfectly elastic reflection of the ball from the boundary such that the angles of incidence and reflection are equal. We add a smooth potential $W(x,y)$; then the equations of motion of the ball (between bouncing off the boundary) have the following form:
$$
\begin{equation}
\begin{cases} \ddot{x}=-W_x, \\ \ddot{y}=-W_y. \end{cases}
\end{equation}
\tag{4.1}
$$
This is a Hamiltonian system on the phase space of the billiard inside an ellipse; its full energy is After a reflection from the boundary, as well as between bouncing, the function $H$ preserves its value. This system is not necessarily integrable; for instance, it is non-integrable for $W=y$. A criterion for the integrability of the Hamiltonian system is due to Kozlov [60].Proposition 10 (Kozlov). A billiard with potential $W$ inside an ellipse admits a first integral of the form $F=\Lambda+f(x,y)$ if and only if the potential satisfies the equation The simplest examples of potential satisfying (4.2) are given by the Hooke potential $W=k (x^2+y^2)$ for $k \in \mathbb{R}$ and by negative powers of coordinate functions $W=\alpha/x^2$ and $W=\beta/y^2$, as well as arbitrary linear combinations of these. The topology of Liouville foliations for such billiard systems was considered by Kobtsev [138]. Some other solutions of Kozlov’s equation (4.2) were found and examined earlier. For instance, Dragović [139] described solutions representable as Laurent polynomials. The functions $V_k(x,y)$ and $W_k(x,y)$ below solve (4.2) for each $k \in \mathbb{N}$, $k \geqslant 2$. For $k=1$ the functions $V_1(x,y)=1/y^2$ and $W_1(x,y)=1/x^2$ are also solutions of (4.2). Dragović proved the following result. Theorem 27 (Dragović). The general solution of (4.2) in the class of Laurent polynomials containing monomial of negative degree has the form of a linear combinations of the functions $V_k$ and $W_k$ such that $V_1(x,y)=1/y^2$ and $W_1(x,y)=1/x^2$ for $k=1$ while for $k \geqslant 2$ Now we find the general solution of (4.2) that is a standard polynomial $W=\sum_{i=0,j=0}^{i+j=n}a_{i,j}x^iy^j$ with real coefficients. In this case Kozlov’s equation transforms into a system of linear equations of the following form for coefficients: where $i>0$, $j>0$, and $i+j\leqslant n+2$. Here we set for $i \in 0,\dots,n+1$. This system has not been solved in the general case, but the following results are known.Lemma 1 (Pustovoitov). The general solution of (4.2) which has the form of a polynomial has the following properties: (a) $a_{i,j}=0$ for $2\nmid i$ или $2\nmid j$, that is, only even coefficients can be distinct from zero; (b) the space of solutions of the form (4.7) has dimension $\lfloor n/2 \rfloor$. It turns out that Kozlov’s equation (4.2) and the polynomial (4.7) have a rather simple form in the elliptic coordinates $(\lambda_1,\lambda_2)$. The transition formulae from $x$, $y$ to $\lambda_1$, $\lambda_2$ are as follows:
$$
\begin{equation}
\begin{cases} x^2=\dfrac{(a-\lambda_1)(a-\lambda_2)}{a-b}\,, \\ y^2=\dfrac{(b-\lambda_1)(b-\lambda_2)}{b-a}\,. \end{cases}
\end{equation}
\tag{4.8}
$$
Coordinate lines are confocal ellipses and hyperbolae from the family (2.1), where $\lambda_1\in[b,a]$ is the parameter of the hyperbola passing through the point satisfying $xy \ne 0$ and $\lambda_2\in(-\infty,b)$ is the parameter of the ellipse through this point. The boundary of the billiard domain has the form $\lambda_2=0$. It can easily be shown that in these coordinates Kozlov’s equation is
$$
\begin{equation}
xy\biggl(\frac{W_1-W_2}{\lambda_1-\lambda_2}-W_{1 2}\biggr)=0,
\end{equation}
\tag{4.9}
$$
where $W_i$ is the partial derivative of $W$ with respect to $\lambda_i$. In addition, functions of the form
$$
\begin{equation}
W=\frac{P(\lambda_1)-P(\lambda_2)}{\lambda_1-\lambda_2}
\end{equation}
\tag{4.10}
$$
solve this equation for each $P \in \mathbb{C}^\infty(\mathbb{R})$. Consider the subclass of solutions for which $P(t)$ is a polynomial of degree $\lfloor n/2\rfloor+1$ with trivial linear part. This subclass has dimension $\lfloor n/2 \rfloor$. After going back to the Cartesian coordinates from elliptic ones we see that all solutions in this subclass are polynomials of $x$ and $y$ with monomials of even degree. For dimension reasons, by and Lemma 1 we arrive at the following result. Lemma 2 (Pustovoitov). The general solution of (4.2) in the form of a polynomial (4.7) has the form (4.10), where $P$ is an arbitrary polynomial of degree $\lfloor n/2 \rfloor+1$. Thus, Pustovoitov proved that any polynomial potential preserving the integrability of the billiard system is coded by a unique polynomial of degree higher than one. For instance, for the Hooke potential mentioned above this polynomial is $P(t)=kt^2$. Pustovoitov also calculated the full system of invariants classifying the Liouville foliations for such billiards. This result can be extended to an arbitrary elementary confocal domain (as it was done for the Hooke potential in [140]). In Fig. 24 we show examples of Fomenko–Zieschang invariants arising in billiard systems with polynomial potentials of low degree. 4.3. Modelling non-degenerate singularities of rank $0$ by billiards with Hooke potentialsIn [35] Fomenko and Vedyushkina stated the problem of modelling an arbitrary non-degenerate four-dimensional singularity of an integrable system by means of a billiard from some suitable class. This is a narural generalization of Fomenko’s Conjecture A on the realization of arbitrary $3$-atoms, which are singularities with non-degenerate points of rank $1$. Such a class was successfully found: it consists of billiard books with Hooke potential which has the same coefficient $k$ on all sheets. Lemma 3. Non-degenerate singularities of any of the four types (centre-centre, centre-saddle, saddle-saddle, and focus-focus) can be modelled by an arbitrary circular or confocal billiard with Hooke potential. For singularities of centre-centre, saddle-saddle, and focus-focus type this follows from an explicit verification of the fact that for the potential $k(x^2+y^2)$ the critical point $(0,0)$ is non-degenerate. The first two types are realized by a billiard system inside an ellipse with Hooke potential such that $k >0$ or $k <0$, respectively, and the last type is realized in a circular billiard. Theorem 28 (Vedyushkina, Kibkalo, and Pustovoitov). An arbitrary non- degenerate semilocal focal singularity of rank $0$ in an integrable Hamiltonian system with two degrees of freedom is topologically modelled by a billiard book of $n$ circular billiards in a disc which are glued in accordance with the permutation $(1,2,\dots,n)$. It is convenient to take for the proof the disc $x^2+y^2 \leqslant 1$, verify that the point $(0,0,0,0)$ is non-degenerate and pass to the polar coordinates. The $r$-coordinate and the component $\dot{r}$ of the velocity are related as follows to the values $H= h$ and $F=r^2\dot{\varphi}=f$ of the first integrals: The domain of possible motion for $H=h$ and $F=r^2 \dot{\varphi}=f$ is determined by the condition which shows that it is an annulus bounded by the boundary of the billiard table from outside, while from inside it is bounded by the circle of radius For $r_0=0$ (that is, for $f=0$ and $h>0$) the domain of possible motion is the whole billiard domain. The bifurcation diagram is shown in Fig. 25; it consists of the isolated point $(0,0)$ and the parabola $h=(f^2+k)/2$. The preimage of this point contains a critical point of focus-focus type, and the preimage of the parabola contains critical circles. The preimage of a small transversal line segment if leafwise homeomorphic to a $3$-atom $A$.Note that after going around the isolated point on the bifurcation diagram a Liouville torus is taken to itself by the monodromy operator. When an attractive Hooke potential is added, the billiard book glued of $n$ sheets of a circular billiard in the disc with permutation $(1,\dots,n)$ models a focal singularity with $n$ points of focus-focus type (Vedyushkina, Kibkalo, and Pustovoitov [67]). An arbitrary non-degenerate semilocal singularity of saddle-saddle or centre-saddle type is modelled by a book constructed algorithmically from this singularity and glued of sheets of type $A_0'$ (Kibkalo; see [65] and [66]). In [65] a limit transition (as the semiminor axis tends to zero) from the system of the flow on an ellipsoid $E^2 \subset \mathbb R^3$ in a central repulsive or attractive Hooke potential to the topological billiard on a table $2 A_2$ with potential $k(x^2+y^2)/2$ was considered: the table is glued of two planar tables bounded by the same ellipse. In [141] Kobtsev investigated this system: he constructed the bifurcation diagrams, determined the number of regular toral leaves in the preimage of a point from each chamber, determined the types of $3$-atoms and calculated the marks on the Fomenko–Zieschang invariants of foliations on isoenergy surfaces. Equilibria of the system correspond to velocity zero and the ball placed at an endpoint of a semiaxis of the ellipsoid. It is easy to verify, by following [6] and considering the form $d^2 H$, that this critical point is non-degenerate and has rank $0$. For pairwise distinct lengths of the semiaxes all critical points are non-degenerate; each endpoint of the semimiddle axis corresponds to one point of centre-saddle type, each endpoint of the semiminor axis corresponds to one point of centre-centre or saddle-saddle type, for an attractive $(k>0)$ or a repulsive $(k<0)$ potential, respectively. The types of equilibria at the endpoints of the semimajor axis depend on $k$ in the converse way. Since the flow on an ellipsoid is a smooth system (with no boundaries or reflections from the boundaries), the Liouville tori containing these non-degenerate semilocal singularities must be non-degenerate semilocal of the same types. Now from the form of the bifurcation diagram we find their types easily: centre-centre $A \times A$, centre-saddle $A \times C_2$, of saddle-saddle $B \times C_2$. Proposition 11 (Kibkalo [65]). Confocal topological billiards with Hooke potential model a non-isolated local singularity of rank $0$ (critical points of rank $0$ must exist in a neighbourhood of the fixed point) of each of the three types centre-centre, centre-saddle, and saddle-saddle, and also model examples of semilocal singularities of each of these types. Note that for a repulsive potential both billiard systems, on billiard tables $A_2$ and $2A_2$ (a billiard inside an ellipse, and one on two such tables glued along the boundary), contain semilocal singularities of types centre-centre $A \times A$ and centre-saddle $A \times C_2$. As regards the case of an attractive potential, the foliation for the billiard system on the table $A_2$, in contrast to $2A_2$, has no singularities homeomorphic leafwise to centre-saddle or saddle-saddle singularities, although in the phase space the two systems have the same set of fixed points projecting onto endpoints of the axes of the ellipse. A similar phenomenon arises in the pair of billiards $A_0'$ and $A_0 \equiv 2A_0'$ on the level $\lambda=b$ of the first integral: the billiard system has a periodic trajectory on the preimage of the $Ox$-axis, and in the case of $A_0'$ the leaf containing it is homeomorphic to a torus (that is, it is not a bifurcation level for the Liouville foliation). On the other hand for the billiard domain $A_0$, or a pair of domains $A_0'$ glued along the $Oy$-axis, the singular leaf is the product of a circle and a figure of eight (whose two ‘loops’ correspond to the two domains $A_0'$, or to the upper and lower halves of $A_0$). As a base domain for modelling centre-saddle and saddle-saddle singularities, we take $\Omega_0:=A_0'$ bounded by an arc of the ellipse $\lambda=0$, an arc of a hyperbola $b<\lambda=h < a$, and the $Ox$- and $Oy$-axes. Theorem 29 (Kibkalo [65]). An arbitrary centre-saddle singularity of type $A \times V$ for some Morse saddle $2$-atom $V$ without stars and of complexity $n$ can be modelled by a billiard book glued of $2n$ domains $\Omega_0$ along boundary segments lying on the $Oy$-axis, with permutation and along boundary arcs of a hyperbola with permutation $\sigma(V) \in S_{2n}$ specified by the $f$-graph for the $2$-atom $V$. In [13] Oshemkov introduced a combinatorial invariant of hyperbolic singularities, which he called an $f_n$-graph. For one degree of freedom this is the $f$-graph for a Morse $2$-atom. For two degrees of freedom the $f_2$-graph is defined by a quadruple $\sigma_1$, $\sigma_2$, $\tau_1$, $\tau_2$ of permutations such that
$$
\begin{equation*}
\sigma_1 \circ \sigma_2=\sigma_2 \circ \sigma_1, \qquad \tau_1 \circ \tau_2=\tau_2 \circ \tau_1,\quad\text{and} \quad \sigma_i \circ \tau_j=\tau_j \circ \sigma_i, \quad i \ne j,
\end{equation*}
\notag
$$
where $\tau_1$ and $\tau_2$ are products of independent transpositions and the orbit of each element has cardinality four. The singularity is determined by an equivalence class of such $f_n$-graphs with respect to two operations: in place of $\sigma_i$ one can take $\sigma_i^{-1}$ or $\tau\sigma_i$. We assign the permutations $\sigma_1$ and $\sigma_2$ to the sides of $A_0'$ lying on the ellipse $\lambda=0$ and the hyperbola, and we assign $\tau_1$ and $\tau_2$ to the sides lying on the $Ox$- and $Oy$-axes, respectively. Theorem 30 (Kibkalo). Given a saddle-saddle singularity of complexity $n$ (that is, one with $n$ non-degenerate equilibria on the singular leaf and satisfying the condition of non-splittability) we glue a billiard book of $4n$ sheets $A_0'$ with permutations $\sigma_1$, $\sigma_2$, $\tau_1$, and $\tau_2$ on the elliptic and hyperbolic arcs and segments of the $Ox$- and $Oy$-axes, respectively. Assume that these permutations define the $f_n$-graph of the singularity in question and let the motion along planar sheets of the book proceed in the field of a Hooke repulsive potential with the same coefficient $k$ on all sheets. Then the Liouville foliation of the resulting complex contains a singularity which is leafwise homeomorphic to the given saddle-saddle singularity. In [66] Kibkalo considered the case of saddle-saddle singularities of complexity $1$ and calculated their circular molecules (recall that in the smooth case the circular molecule is a distinguishing invariant for saddle-saddle singularities of complexity $1$ and $2$). In the general case, according to Lazutkin’s [142] smoothing results, for such a system we can introduce a smooth symplectic structure in a neighbourhood of a leaf containing points of saddle-saddle type. Each quadruple of domains $A_0'$ glued in accordance with the permutations $\tau_1$ and $\tau_2$ corresponds to a single domain $A_0$ bounded by the same ellipse and hyperbola. For such a planar table all trajectories close to the singular leaf (containing saddle-saddle equilibria) are transversal to boundary arcs at points of reflection. This means that the singularity of the billiard system has the saddle-saddle type indeed, after which we can calculate its $f_n$-graph and verify that it coincides with the invariant of the singularity we model. 4.4. Integrable magnetic topological billiardsConsider a planar billiard system endowed with a constant magnetic field orthogonal to the plane of the table. In Cartesian coordinates the equations describing the motion of the billiard ball between bouncing off the boundary have the form
$$
\begin{equation}
\begin{cases} \ddot{x}=-b\dot{y}, \\ \ddot{y}=b\dot{x}, \end{cases}
\end{equation}
\tag{4.13}
$$
where $b>0$ is the coefficient depending on the charge of the ball and the value of the magnetic induction. The Hamiltonian of this system is equal to the kinetic energy: $H=(\dot x^2+\dot y^2)/2$. Such billiards, for domains of various shape, have been considered from various points of view. For instance, Robnik and Berry [143] considered a magnetic billiard in an elliptic domain from the standpoint of the behaviour of trajectories and the stability and character of dynamics. They showed that there exists levels of $H$ on which the dynamics is chaotic. That the magnetic billiard inside an ellipse is not integrable for magnitudes $\varepsilon$ of the magnetic field was proved by Kozlova [144]. To within $\varepsilon^2$ this system is equivalent to the motion of a particle bouncing off an ellipse rotating with constant angular velocity $\varepsilon/2$ about its centre. Mironov and Bialy [62] established a criterion for the integrability of a magnetic billiard in a simply connected plane domain (and subsequently, also for similar domains on a sphere or a Lobachevsky plane). Namely, the following result holds. Theorem 31 (Bialy and Mironov). Let the boundary of a magnetic billiard domain be connected, convex, and distinct from a circle. Then this billiard is not algebraically integrable for any value of the magnitude of the magnetic field $b$, with the possible exception of a finite number of values. On the other hand a magnetic billiard bounded by a circle has a first integral independent of the Hamiltonian. In the Cartesian coordinates it has the form
$$
\begin{equation}
F=\dot{x}^2+\dot{y}^2+b^2(x^2+y^2)-2b(x\dot{y}-y\dot{x}).
\end{equation}
\tag{4.14}
$$
Note also that the function $F$ is independent of the radius of the boundary circle. Moreover, $F$ is also preserved under outer reflections of a ball from the boundary. Hence the magnetic billiard in a domain bounded by two concentric circle is also integrable with the same additional first integral. The first integrals $H$ and $F$ are geometric. Namely, the system of equations (4.13) defines the (anticlockwise) motion of the billiard ball along a circle of radius $L= \sqrt{2H}/b$. These circular trajectories are called Larmor circles. The centre of a Larmor circle lies at distance $R=\sqrt{F}/b$ from the centre of the billiard table. In Fig. 26 we show the trajectory of a billiard ball, segments of which lies on Larmor circles of equal radii, with centres lying at the same distance from the centre of the table. It is not difficult to see that all trajectories of the billiard ball that correspond to some levels $H=h$ and $F=f$ lie inside the intersection of the billiard table and the annulus bounded by the circles of radii $|R-L|$ and $R+L$. Moreover, each point in this domain lies on some trajectory. Hence this is the domain of possible motion. Note that there are two velocity vectors corresponding to each interior point of this domain, because there are precisely two Larmor circles through this point. This yields the following result. Lemma 4. The preimage of an arbitrary domain of possible motion of a magnetic billiard under the projection $p$ (that is, the corresponding leaf of the Liouville foliation) is homeomorphic to a $2$-torus in $Q^3$, and the foliation in a neighbourhood of this torus is trivial. Next we investigate the global structure of Liouville foliations for magnetic billiard tables bounded by one or two circles. Fix a value $H=h$ distinct from zero and let us monitor how the domain of possible motion changes as $F$ varies from the minimum value $f_{\min}$ to its maximum $f_{\max}$ (see Fig. 27). By Lemma 4 each value of $F$ distinct from the minimum and maximum corresponds to a Liouville torus, and the minimum and maximum values of this integral correspond to singular leaves of the atom $A$. Thus, the rough molecule of a magnetic billiard has the form $A$–$A$. To calculate the marks, as in the case of planar billiards, we consider the projections of cycles in admissible bases onto the billiard table. For the circular billiard in a disc we have $r=0$ and $\varepsilon=1$, while for the circular billiard in an annulus we have $r=\infty$ and $\varepsilon=1$. Now consider a topological billiard glued of round discs and annuli. Such a billiard table can be homeomorphic to one of four orientable manifold: a cylinder or a torus (when all sheets are annuli), a disc (when there is a unique sheet homeomorphic to a disc), or a sphere (when there are two such sheets). On each sheet motion is defined as on the corresponding planar magnetic billiard, with the same magnitude of the field $b$ on all sheets. This system has the same first integrals $H$ and $F$ as planar magnetic billiards. Fix a value of energy $H$ (or, equivalently, the radius $L$ of a Larmor circle). We describe an algorithm for the construction of the Fomenko–Zieschang invariant of a magnetic billiard by considering a table homeomorphic to a cylinder for example (in the other three cases there are similar algorithms). Step 1. We number the gluing edges of the billiard table in accordance with the elementary domains (sheets) containing them, so that the components of the boundary of a sheet must be numbered by consecutive positive integers. Step 2. In the $Oxy$-plane we construct a polygonal curve (the line $K_0$ in Fig. 28, (a)) as follows: we join successively the points with coordinates $(i,R_i)$, where $R_i$ is the radius of the $i$th part of the boundary. Then we reflect the resulting graph in the $Ox$-axis (the curve $M$ in Fig. 28, (a)). In other words the polygonal lines $K_0$ and $M$ form in combination the profile of a topological billiard (its cross-section by a plane containing the symmetry axis of the billiard). Finally, we reflect the part of $K_0$ lying above the line $y=L$ in this straight line (the polygonal curve $K$ in Fig. 28, (a)). Step 3. We partition the domain between $K$ and $M$ into horizontal line segment and contract each segment to a point. In the resulting graph we assign the index $A$ to all free vertices. Step 4. The other vertices of the resulting graph correspond to some local minima on the curve $K$ and some local maxima on the curve $M$ (all of which lie on some horizontal line segment). To this segment we assign the sequence of minima and maxima ordered from left to right. It is coded by the numbers of successive minima or maxima (from left to right; $\tau_1$ successive minima, then $\tau_2$ successive maxima, and so on); for the segment shown in Fig. 28, (c), this code has the form $\tau=(2,2,1,1)$. To such a vertex we assign the coded atom $B_\tau$. Step 5. The atom denoted by $B_\tau$ is constructed from the $2$-atom $B_n$, where $n=\sum{\tau_i}$, as follows. We cut $B_n$ transversally into $|\tau|$ connected parts so that the $i$th part contains precisely $\tau_i$ critical points. Then we glue these parts back with twist (Fig. 28, (d)). From the $2$-atom thus constructed we obtain a $3$-atom by taking its direct product with a circle. Note that the class of $3$-atoms $B_\tau$ coincides precisely with the class of atoms $V_n^{\eta_1,\dots,\eta_n}$ (so-called atoms with pluses and minuses) which arose in [86] in the description of flows on surfaces of revolution in a magnetic field. We changed the notation just to make it more convenient in the context of the particular problem under consideration. Theorem 32 (Pustovoitov). The graph with indexed vertices (Fig. 28, (b)) produced by the algorithm is the rough molecule of the cylindrical topological magnetic billiard for a fixed level of energy $H$. Theorem 33 (Pustovoitov). Marks in the Fomenko–Zieschang invariant of a topological magnetic billiard glued of discs and annuli bounded by concentric circles have the following form: 1) the mark $r$ is equal to zero on all edges incidents to atoms $A$ and to infinity on all other edges (so that there is a unique family); 2) the mark $\varepsilon$ is equal to $-1$ on all edges connecting two saddle atoms corresponding to different directions of motion about the origin; otherwise it is equal to $+1$; 3) the mark $n$ on the unique family is equal to zero for billiards homeomorphic to a cylinder or a torus, to $\pm1$ in the case of a billiard in a disc, and to $\pm2$ in the case of a billiard on a sphere. Thus, Pustovoitov performed a complete analysis of Liouville foliations for magnetic topological billiards. An example of a possible Fomenko–Zieschang invariant is shown in Fig. 29. 4.5. Multidimensional billiards and a generalization of the Jacobi–Chasles theoremSimilarly to Birkhoff’s argument that a billiard inside an ellipse is integrable, one can show that multidimensional billiards in domains bounded by confocal quadrics are integrable. The number of functionally independent involutive first integral coincides with the dimension of the billiard. For example, given a billiard in a three-dimensional domain $\Omega\subset\mathbb{R}^3$ bounded by confocal quadrics, links of trajectories lie on straight lines that are simultaneously tangent to a pair of confocal quadrics. Apart from the energy, two other functions are preserved on trajectories [61], [68]. The topological properties of the Liouville foliations for such billiards were presented by Dragović and Radnović [100] and Belozerov [68]. Fairly recently Kibkalo considered the question of the integrability of the geodesic flow on the intersection of several confocal quadrics of different types. He showed that the geodesic flow on the intersection of $n-2$ quadrics in $\mathbb{R}^n$ is a completely integrable system and indicated a first integral explicitly in the elliptic coordinates. The metric on the surface can be reduced to the Liouvillean form. Thus, he constructed a rather ample class of two-dimensional domains that, just as elementary plane domains, can be used to construct integrable topological billiards and billiard books with non-plane metric. It turns out that this result can be generalized by considering the geodesic flow on the intersection of an arbitrary number of non-degenerate confocal quadrics. Theorem 34 (Belozerov). Let $Q_1,\dots,Q_k$ be non-degenerate confocal quadrics of pairwise different types in $n$-dimensional Euclidean space, and let $Q=\bigcap_{i=1}^k Q_i$. Then 1) the geodesic flow on $Q$ is completely integrable, with $n-k$ first integrals which are quadratic in momenta; 2) tangent lines to a fixed geodesic at all points on it are also tangent (apart from $Q_1,\dots,Q_k$) to $n-k-1$ other quadrics confocal with $Q_1,\dots,Q_k$, which are the same for all points on this geodesic. Remark 9. Geodesics on an intersection of several confocal quadrics are, generally speaking, not geodesics on any of the quadrics $Q_1,\dots,Q_k$. Thus, Theorem 2 is not a consequence of the classical Jacobi–Chasles theorem. Next, $Q$ is homeomorphic to a product of $k$ spheres; their dimensions are equal to the numbers of non-fixed elliptic coordinates placed in between consecutive fixed ones. A family of confocal quadrics in $\mathbb{R}^3$ is the set of quadrics
$$
\begin{equation*}
(b-\lambda)(c-\lambda)x^2+(a-\lambda)(c-\lambda)y^2+ (a-\lambda)(b-\lambda)z^2=(a-\lambda)(b-\lambda)(c-\lambda),
\end{equation*}
\notag
$$
where $a>b>c>0$ are fixed numbers and $\lambda$ is a fixed parameter. If a quadic in this family has parameter $a$, $b$, or $c$, then it is said to be degenerate, otherwise it is said to be non-degenerate. Given a family of confocal quadrics in $\mathbb{R}^3$, by a three-dimensional billiard table we mean a compact subset of $\mathbb{R}^3$ with non-empty interior that is bounded by a finite number of smooth faces lying on quadrics in this family, with dihedral angles at corners of the boundary equal to $\pi/2$. Consider the following dynamical system: a unit point mass (ball) moves in the interior of the billiard table along straight lines with constant modulus of velocity, bouncing off the boundary absolutely elastically. Since all dihedral angles at the boundary are equal to $\pi/2$, the reflection at corners can be extended by continuity. We call this dynamical system a three-dimensional billiard. Now we describe the phase space of this system. Let $\Omega$ be a billiard table traversed by a billiard ball. Let
$$
\begin{equation*}
\widehat{M^6}=\{(x,v)\mid x\in \Omega,v\in T_{x}\mathbb{R}^3\}.
\end{equation*}
\notag
$$
Then the phase space of this system is the manifold $M^6=\widehat{M^6}/{\sim}$, where $\sim$ is the following equivalence relation: $(x_1,v_1)\sim(x_2,v_2)$ if and only if one of the following conditions holds:
Note that the system under consideration is an integrable Hamiltonian system in the piecewise smooth sense. In our case the Hamiltonian is equal to the modulus of the velocity vector: $H(x,v)=\|v\|$. Two further integrals which are functionally independent of $H$ are the parameters of the confocal quadrics in this family that are simultaneously tangent to all straight-lined trajectories of the ball. They exist by the Jacobi–Chasles theorem. In fact, consider the geodesic flow on a three-dimensional ellipsoid in $\mathbb{R}^4$, and let the semiminor axis of the ellipsoid tend to zero. Then this geodesic flow turns to the three-dimensional billiard inside an ellipsoid in $\mathbb{R}^3$. First integrals of this system are equal to the parameters of the two quadrics confocal to the ellipsoid and tangent simultaneously to all straight-lined trajectories of the ball. Let these parameters be $\Lambda_1$ and $\Lambda_2$. Note that a trajectory cannot be tangent to two confocal ellipsoids or two confocal hyperboloids of two sheets simultaneously: confocal ellipsoids are disjoint and ar convex surfaces, and the same holds for confocal hyperboloids of two sheets. Hence we can assume that ${\Lambda_1\geqslant\Lambda_2}$, $\Lambda_1\in[c,a]$, and $\Lambda_2\in[x_0,b]$, where $x_0$ is the least parameter of the ellipsoids forming the boundary of the billiard table. We can show that the functions $H$, $\Lambda_1$, and $\Lambda_2$ commute pairwise with respect to the standard Poisson bracket for $x\in \operatorname{Int}(\Omega)$. As the investigation of dynamical systems with three degrees of freedom is much more complicated than the investigation of ones with two degrees of freedom, consider a ‘simple’ equivalence relation on the set of integrable Hamiltonian systems with three degrees of freedom. Definition 19. Let $v_1$ and $v_2$ be Liouville integrable Hamiltonian systems. Let $B_1$ and $B_2$ be the bases of the Liouville foliations for $v_1$ and $v_2$, respectively. We say that $v_1$ and $v_2$ are weakly equivalent if there exists a homeomorphism $\varphi\colon B_1\to B_2$ between the bases of Liouville foliations such that for each $x\in B_1$ the leaves of the foliations for $v_1$ and $v_2$ corresponding to $x\in B_1$ and $\varphi(x)\in B_2$ are homeomorphic. Note that this equivalence relation is much weaker that rough Liouville equivalence. Furthermore, the relation of weak equivalence can be limited to Liouville foliations on isoenergy surfaces. The problem of the classification of three-dimensional billiards with respect to weak equivalence was solved by Belozerov [68]. For the first time the Liouville foliation for a three-dimensional billiard was described in similar terms by Dragović and Radnović [100] in the case of a three-dimensional billiard inside an ellipsoid. Here we describe one billiard system considered by Belozerov. Consider a three-dimensional billiard table bounded by an ellipsoid and two fragments of a confocal hyperboloid of two sheets. Assume that the boundary ellipsoid has the equation
$$
\begin{equation*}
\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}=1,\qquad a>b>c.
\end{equation*}
\notag
$$
Such a table is shown in Fig. 30, (a). Fix the energy level $H=1$ and consider the restriction of the moment map to the isoenergy surface $Q^5$. For the first integrals $\Lambda_1$ and $\Lambda_2$ (the parameter of the caustics) we have $\Lambda_1\geqslant\Lambda_2$, $\Lambda_1\in[c,a]$, and $\Lambda_2\in[0,b]$. In Fig. 30, (b), the image of the moment map is shaded and the bifurcation diagram $\Sigma$ is drawn by continuous black lines. Note that a circular molecule arises at the point $(b,c)$, which is a topological invariant of the singularity corresponding to the point of intersection of two line segments (this invariant is non-trivial when both segments are parts of the bifurcation diagram). Recall that the circular molecule of a point in the image of the moment map for a system with two degrees of freedom is the molecule (Fomenko invariant) of the Liouville foliation on the $3$-dimensional boundary of an invariant $4$-dimensional neighbourhood of the preimage of this point. In our case the system has three degrees of freedom, but we have fixed $H=1$. Thus, it is natural to mean by the circular molecule of ${(b, c)}$ the invariant of the foliation on the $4$-dimensional boundary of the $5$-dimensional neighbourhood of the preimage of this point. For the billiard table under consideration the circular molecule of the singularity corresponding to $(b,c)$ is as in Fig. 31. Vertices of this molecule correspond to the Cartesian products of $ 3$-atoms and a circle. For this table one can fully describe the topology of the Liouville foliation of a small neighbourhood of a point in the cross. Since our billiard table is remarkably foliated by confocal hyperboloids of two sheets, a smooth neighbourhood of the leaf corresponding to the point $(b,c)$ is homeomorphic to the Cartesian product of a circle $S^1$ and the complex $K^4$ that is a neighbourhood of the leaf containing a saddle-saddle singular point of the Liouville foliation for the two-dimensional billiard with a Hooke potential inside an ellipse. It follows from Kibkalo’s results that $K^4\simeq(B\times C_2)/\mathbb{Z}_2$. Thus, a small neighbourhood of the leaf over $(b,c)$ if homeomorphic to $S^1\times(B\times C_2)/\mathbb{Z}_2$. The authors are grateful to V. A. Kibkalo for a number of valuable comments, which contributed to the improvement of the quality of the text. 
Citation in format AMSBIB
\Bibitem{FomVed23} |
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