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This article is cited in 3 scientific papers (total in 3 papers) Evolutionary force billiards A. T. Fomenko, V. V. Vedyushkina Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 5, DOI: https://doi.org/10.4213/im9149 
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    § 1. Introduction. A force (evolutionary) billiard. A visual description1.1. Invariants of Liouville foliations on $Q^3$ and their realization by billiardsQualitative analysis of dynamical systems turned out to be an important approach to researching into phase topology of such systems (Smale [1]). For integrable Hamiltonian systems, the phase space of the system has a structure of the Liouville foliation, i.e., the space is partitioned into regular tori and singular fibers. In the case of two degrees of freedom, in [2]–[6] Fomenko, representatives of his school of thought and co-authors built the theory of topological classification of such systems, restricted to a three-dimensional nonsingular level $Q^3$ of energy $H=h$, using the Fomenko–Zieschang invariants (finite graphs with numerical marks). It is described in detail in the monograph by Bolsinov and Fomenko [7]. It turned out that systems of different nature are Liouville equivalent to each other in some non-singular zones of energy, i.e., the closures of almost all their solutions have the same structure. Naturally a question about realization (or modeling) of the Liouville foliation of a more complex (in a sense) system using a simpler one arose. The class of integrable billiards turned out to be very promising for such a task for a number of reasons. Firstly, special motion modes of such systems are visualized better (for example, when projected onto a billiard table), in contrast to classical integrable systems in mechanics. Secondly, this class allows a variety of generalizations and extensions, preserving system integrability. They include both new ones, such as gluing a locally flat table which is a complex of flat domains along common boundary arcs suggested by Vedyushkina [8], and previously known ones: adding a suitable potential, a constant magnetic field, or a constant curvature metric. Two key classes of flat integrable billiards (without potential) – confocal and circular billiards – are set on tables bounded by arcs of confocal quadrics or arcs of concentric circles and sections of their radii. In a series of papers [9]–[12] by Vedyushkina and Fomenko, use of integrable billiards, for instance, topological billiards and billiard books, allowed realization (in terms of Liouville equivalence) of many known integrable Hamiltonian systems with two degrees of freedom (hereinafter referred to as IHS). However, each time that was achieved “separately” on one three-dimensional isoenergy surface. This is related to the fact that topological billiards [8], [13], [14], as well as billiard books [15], [16], introduced by Vedyushkina, the energy of a billiard ball (material point) is just a scale parameter. This means that changing it does not change the topology of the Liouville foliation on three-dimensional isoenergy surfaces. At the same time, most integrable systems in geometry and physics “live”, being parameterized by the value of energy (the Hamiltonian), simultaneously on a variety of three-dimensional isoenergy surfaces, corresponding to different energy levels, and other system parameters. Systems with two degrees of freedom “live” on four-dimensional symplectic phase manifolds $M^4$. These four-dimensional manifolds are divided into fibers on constant energy surfaces. For almost all values of energy isoenergy surfaces are regular and three-dimensional. In case of singular values of energy they are generally not three-dimensional manifolds, i.e., they are singular fibers of the foliation $M^4$ into constant energy levels. At the same time, changing the energy zone (transition through a singular level) can lead to a change in topology of three-dimensional isoenergy surfaces. Therefore, the integrable system “evolves”, rearranges itself while energy increases. The energy value is an important parameter on which many properties of the system and its behavior depend. We note that many classical dynamical systems allow a task based on six-dimensional Lie algebras. In this case the four-dimensional symplectic leaf (with IHS on it) can be chosen by fixing a non-singular pair of values for Casimir functions of the Lie–Poisson bracket (for a number of systems matching the geometric integral and the area integral). To date, behavior of many integrable systems in geometry, dynamics and mathematical physics, as well as their analogues in Lie algebras has been described using the Fomenko–Zieschang invariants or the Fomenko invariants [17] (a graph without numerical marks) of their Liouville foliations on non-singular isoenergy three-dimensional surfaces $Q^3$. They include classical dynamical systems: integrable Euler and Lagrange [7], Kovalevskaya [18], Clebsh [19], Steklov and Sokolov [20], Kovalevskaya–Yahya cases [21], [22], and their analogues in Lie algebras [23]–[26]. Non-compact systems, their invariants (see [27]–[30]) and possibility of their realization by billiards [11] are being actively studied. For example, in the article [30] integrable systems with two degrees of freedom with incomplete flows are studied. It describes the Liouville foliation near a regular fiber, with accuracy up to fibered symplectomorphism. Classification of geodesic flows with integrals of powers 1 and 2 followed by calculation of their invariants (see [31]–[33]) turned out to be an important area of research. The class of such integrable geodesic flows in the case of orientable surface (a sphere or a torus) was fully modeled Vedyushkina and Fomenko in [34]. We would also like to note a recent work by Kantonistova [35], Timonina [36], Kudryavtseva and Oshemkov [37], who studied the topology of integrable systems on surfaces of revolution. It was possible to realize a large class of integrable systems (with restriction on the third energy level) and their topological invariants using suitable billiards. In [38] Fomenko formulated the fundamental conjecture about realizing integrable systems and their invariants using billiards. A number of its provisions have been fully proved to date: billiards allowed realizing nondegenerate Bott 3-atoms [15], [16], coarse molecules [39] (bases of Liouville foliations), numerical invariants of IHS (a local version of the hypothesis by Fomenko, [40], [41]). In addition, Vedyushkina constructed billiard books realizing Q 3 from some homeomorphism classes [42]. We would like to remind that the class of isoenergy manifolds in IHS matches (see [7]) the well-known class of graph manifolds introduced by Waldhausen [43], [44]. Item $C$, in Fomenko’s conjecture is its strongest section, which assumes the possibility of realizing an arbitrary Fomenko–Zieschang invariant with the help of a billiard (from a suitable class). This item has not yet been fully proved. Nevertheless, this has been done for many classes of Liouville foliations for systems in dynamics and geometry [9], [11], [45] and [46]. For this purpose various new classes of billiards: circular and confocal billiards, topological billiards glued from them and billiard books, billiard systems in the Minkowski metric [47], billiards in a magnetic field [46] or in the field of action of potential forces [48]–[50], geodesic flows on quadrics [51], including the Hooke potential field on an ellipsoid [52]. Issues related to integrability and properties of billiards or geodesic flows, including those with the addition of various structures (a potential, a magnetic field, the Minkowski metrics), are described in detail in monographs [53]–[55]. In recent works by Glutsyuk [56], Mironov and Bialy [57]–[59] various versions of the Birkhoff conjecture were proved in the case of polynomial integrability. It turned out that confocal and circular billiards are indeed the only (given some conditions and clarifications) integrable billiards with a smooth boundary. This effect occurs both when a constant magnetic field is added (suitable only for circular billiards), and in case of transition to the Lobachevsky sphere and plane, simply connected spaces of constant curvature. In the works by Sorrentino and Kaloshin [60], [61] a local version of the Birkhoff conjecture was proved. Therefore, the transition to billiards on cell complexes (billiard books and topological billiards) proposed by Vedyushkina made it possible to significantly expand the class of systems under consideration in the presence of fundamental restrictions related to the border form of the billiard table for its integrability. 1.2. Evolutionary force billiardsNaturally a question arose: is it possible to define a new class of billiards that realize IHS “not in parts”, but as a whole on the entire 4-dimensional phase manifold $M^4$? In other words, we aim to realize this Hamiltonian system in its evolution, as if it was “sweeping in time”, where the energy of the system performs the role of time. That is, in such a way that the changing billiard realizes the system “as a whole”, on the entire $M^4$ at once. Therefore, changes in the system occurring with increase of its energy would be shown. Remind that, based on an integrable system on a symplectic leaf $M^4$, it is possible to construct a “code”, i.e., a sequence of Fomenko–Zieschang invariants for each regular energy zone of the system. Based on locally flat billiards, we will search for a new class, suitable for realizing the IHS “code”. Therefore, the above-mentioned class should realize the topology of Liouville foliations (solution closures) of the system in all non-singular energy zones. It was found out that such a class of billiards does exist. At the same time, we will not introduce potentials or impose additional conditions on the four-dimensional topology of the billiard system. This is related to the fact that adding a potential to a billiard or a geodesic flow generates a system not only with a non-trivial “code”, but also with other purely four-dimensional invariants. The complete classifying invariant IHS on $M^4$ with some additional requirements was proposed by Zung [62], but the authors do not know examples of its calculation for specific systems from applications. The simplest examples of four-dimensional invariants are represented by local and semilocal topological invariants of four-dimensional system features. Most IHS from applications contain such four-dimensional features, both non-degenerate (the classification was proposed by Zung, see [63]) and degenerate (the case of parabolic features is described in [64]–[67]). The study of IHS features and their invariants is a separate area of science. Recent results in this domain were obtained and presented in [63], [64], [68]–[72]. In [73] the invariant (a circular molecule) of some saddle singularities is realized as a billiard book with the Hooke potential. We note that adding a potential, for example, to a locally flat billiard, makes the dynamics of the system more complex: the links of the trajectory of general types cease to be rectilinear segments (between hits of the ball on the border). We will try to realize IHS with billiards with classical movement of a material point along geodesics of a locally flat metric or Riemannian metrics with an integrable geodesic flow. The answer to the above question was positive: Fomenko introduced a new class of billiards called “evolutionary billiards”. Sometimes we will also use the term “force billiards”. The term “evolutionary” means that the type of billiard table depends on a real parameter, conventionally called energy. The term “force” indicates the force with which the billiard ball (material point) hits against the wall (boundary) of the billiard table. As far as we know, in the previously studied mathematical models of billiards this force was not taken into account. More precisely, in the absence of external forces, it was believed that the speed of the billiard ball is constant in terms of absolute value and, for simplicity, was assumed to be equal to one. As a result, the modulus of speed (the first integral of the system) turned into a simple scale parameter that had no significant impact on the behavior of the system. For instance, its change had no effect on the topology of solutions, for example, on the geometry of bifurcations of integral trajectories. We consider the terms “force billiards” and “evolutionary billiards” to be equivalent, and we will use them interchangeably. In such systems, as energy increases, a billiard changes its topology and the laws of “reflection and refraction” on its borders and “ribs”. We assume that the force with which the ball hits the boundary of the two-dimensional billiard table is determined by the ball speed. The proposed idea of a force billiard is that with a change in the speed of the ball (impact force), both the geometry of the billiard table and the ball’s law of reflection and refraction will change. We can assume that a two-dimensional table is not necessarily flat or locally flat (in the sense of the Euclidean metric). In our works we have already considered billiard tables that are two-dimensional (or multidimensional) Riemannian manifolds across which the point moves along geodesic trajectories, being reflected from the boundaries (“walls”). But currently, for simplicity, we consider locally flat billiards. The idea of the evolutionary billiard is to consider the billiards that depend on the parameter (energy, time) and change, rebuilding themselves according to some “natural” rules. The concept of “natural character” can be formalized in different ways. The fact that on this basis the authors succeed in discovering previously unknown links between various integrable systems is an important argument for the “natural character” concept introduced below. 1.3. Main resultsIn this section we briefly describe the main results of this work. Details will be provided in the next sections. The first resultIt turned out that evolutionary (force) integrable billiards realize (in the sense of Liouville equivalence) some important and well-known (in applications) Hamiltonian systems “entirely”, i.e., immediately on the entire phase symplectic manifold $M^4$ (except for, perhaps, singular isoenergy three-dimensional surfaces). In other words, immediately on all regular isoenergy three-dimensional surfaces. That is, with growth of energy $h$ of a material point a billiard table changes its topology in a quite simple and visual way, and (also visually) the laws of reflection and refraction change on the edges of the billiard (on its “slopes”). At the same time, three-dimensional levels of constant energy of an evolving billiard gradually change (also in a simple way). As a result, the integrable and evolving billiard system “living” on these successively changing energy levels realizes the Hamiltonian system from geometry, topology and mathematical physics at all its energy levels step by step. As vivid examples, we “completely” realized the Euler and Lagrange systems. The case of Goryachev–Chaplygin–Sretensky well known in dynamics of a heavy rigid body was also realized on a suitable interval of energy values (currently its realization has not been completed). The second resultIt turned out that billiard realization reveals “confocal quadrics” as hidden parameters in the Euler case, and in the Lagrange case, “hidden concentric circles”. Eventually, it is the natural and simple deformation of confocal quadrics in a circle (when foci merge) that “turns” a full set of Liouville foliations for the Euler case into a complete set of Liouville foliations for the Lagrange case. We would like to remind that the Euler case is integrable using a quadratic integral, and the for the Lagrange case a linear integral is used. Such “conversion” of a quadratic integrable system into a linear integrable one is an interesting fact. We will state that, in the indicated sense, the Euler system and the Lagrange system are “billiard-equivalent”. Remark 1. In [52], by adding the Hooke potential to the geodesic flow on a triaxial ellipsoid, all three different types of motion of the Euler system were realized. Each of them was implemented in a particular energy zone. However, in contrast to the evolutionary billiard constructed by us, they were not obtained directly from each other during evolution of the system (i.e., with increasing energy). The point is that the flow on an ellipsoid with a potential also has other energy zones, with more complex Fomenko–Zieschang molecules. There were no such molecules in the Euler system. § 2. Definitions and task setting. Main properties of evolutionary billiards2.1. Evolutionary billiard supportThe following definitions were introduced by Fomenko. Definition 1. As a support (i.e. model space) $X$ for an evolutionary force billiard $X(h)$, we consider a finite two-dimensional complex $X$ containing vertices, edges, and two-dimensional closed domains-leaves $L_i$, homeomorphic to closed simply connected domains of the Euclidean plane, i.e., two-dimensional disks. Each edge is considered to be homeomorphic to a closed segment, i.e., its edge is two vertices. The boundary of each closed leaf $L_i$ consists of a finite number of edges. Further we will set a locally flat Euclidean metric on each such sheet of a billiard support. We will require that the angles between the edges are equal to $\pi/2$. If two sheets have a common boundary edge along which they are glued, then we assume that this gluing is isometric. That is, the boundary edge of one sheet is glued isometrically with the boundary edge of the neighboring sheet. Therefore, the complex $X$ is obtained by gluing locally flat billiard sheets along some boundary edge spines. We will call the obtained two-dimensional complex $X$ the evolutionary billiard support. Remark 2. We could consider the billiard system not to be integrable in any way. Of course, integrable billiards are of primary interest to us. However, many issues are also interesting in relation to non-integrable billiards. Remark 3. In the theory of flat integrable billiards, there are planar multiply connected billiard sheets (for example, those bounded by two confocal ellipses), as well as billiards whose boundaries are regular curves without kinks (a billiard in an ellipse). However, such billiards are easily partitioned into billiards homeomorphic to disks, the boundary of which consists of segments joined at right angles. Later we will also consider billiards whose sheets can be multiply connected (but easily partitioned into discs, so dynamics remain unchanged). For instance, in our work we will need billiards bounded by concentric circles. Remark 4. As we stated above, instead of a locally flat metric on evolutionary billiard support sheets, the Riemannian metric can be set. We give such an example below. Many interesting questions arise in this area. We denote the modulus of the velocity vector of a material point (its energy) by $H$, and its particular value will be denoted by $h>0$. 2.2. An evolutionary (force) billiard as a dynamical system on the changing billiard tableDefinition 2 (evolutionary billiard support and changing billiard states). This definition includes the following four points. 1. We call a connected two-dimensional locally flat cell complex $X$ (“table”) described in definition 1 an evolutionary billiard support. For example, billiard books $X$ (see [16], [42]) will be an important class interesting for applications. We recall that such billiards define integrable systems with two degrees of freedom. 2. For each value $h$, we consider a closed cell subcomplex $X(h)$, which is not necessarily connected, in a connected support $X$. We consider that energy $h$ changes from 0 to infinity. We call $X(h)$ an evolutionary (force) billiard state corresponding to the value $h$. When the parameter $h$ changes, the state $X(h)$ will change. We assume that the complex $X$ is an association of all evolutionary billiard states. That is, all states “live” inside the ambient support complex $X$, continuously change with the growth of $h$ in some way, and their association exhausts the entire support $X$. Moreover, as $h$ increases, the state of $X(h)$ increases. That is, if $h_1<h_2$, then $X(h_2)$ contains or is the same as $X(h_1)$. In other words, as $h$ grows, the state $X(h)$ “absorbs” all previous states, see Fig. 1 below. This is natural, since the parameter $h$ is used in the meaning of energy. With increase in the ball energy the billiard area increases, i.e., the ball penetrates into a large area in the support complex. However, some walls which used to be impenetrable for the ball, now become penetrable, or they are smoothly deformed (moved aside), affected by ball impact. Therefore, the area of the state $X(h)$ grows monotonously. On the variation interval of $h$ we single out a finite number of values $h=1$, $h=2$, $\dots$, $h=N$, which we call singular once, while the remaining values are said to be regular, see Fig. 2. 3. Edge spines of states $X(h)$ are the arcs of confocal quadrics or segments of focal lines. As part of state boundaries $X(h)$, there can also be circles, which, at critical values of energy, can degenerate into points. 4. An evolutionary (force) billiard on the support $X$ is a dynamical system set by the motion of a material point (a billiard ball) along the segments of geodesics inside the leaves $L_i$ with a constant velocity modulus equal to $h$, with its law of reflection and refraction $Z(h)$ on the boundary of each billiard sheet, i.e., either on the edge spine or on the boundary circle. Remark 5. Sometimes it is convenient to consider the support $X$ and the state $X(h)$ with accuracy up to homeomorphism, i.e., temporarily ignoring the existence of a locally flat metric on billiard sheets. This allows a clearer representation (for example, depicting on two-dimensional models in the three-dimensional Euclidean space) of the support topological properties and billiard states. A specific example is shown in Fig. 1. It turns out that it emerges when the Euler case in rigid body dynamics is analyzed. Here, the locally flat support $X$ is homeomorphic to a two-dimensional ellipsoid, and the states $X(h)$ are homeomorphic to smoothly deformed areas on the ellipsoid. Remind the informal definition of a billiard book book (for more details, see [16] by Vedyushkina and Kharcheva). Consider a complex $X$ consisting of two-dimensional cells-sheets. Recall that we required for each cell to be a region on a plane bounded by a piecewise smooth curve with all angles equal to $\pi/2$. These cells are glued to each other along one-dimensional edge spines. We now number the book sheets. Every one-dimensional edge matches a cyclic permutation from the group $S(k)$, where $k$ is the number of book sheets converging on a particular edge. At the same time, permutation is recorded in such a way as to keep billiard sheet numbering uniform. We project $X$ onto the plane. Consider a multitude of boundaries of all billiard sheets. Let two boundaries form an angle on the plane. We require that in their preimage the permutations corresponding to them commute. Here, permutation in the preimage of a smooth boundary arc means permutation consisting of united cyclic permutations of all spines projected onto the given arc. Then the motion on the obtained complex is defined as follows. A material point moving on the sheet $i$ is transferred to the sheet number $\sigma(i)$ after hitting a boundary, where $\sigma$ – is the permutation assigned to the given boundary. The commutation condition ensures that the reflection when hitting an angle is correct. In this case the sheet number is determined by composition of permutations assigned to the arms of the given angle. 2.2.1. Laws of reflection and refraction, penetrable and impenetrable edgesThe next four points describe the laws of reflection and refraction. 1. The law of reflection and refraction generally depends on the parameter $h$ and on the edge of gluing $r$ on the billiard sheet boundary in the state $X(h)$. Therefore, we will write it as $Z(h,r)$. This means a task for a cyclical permutation from a permutation group $S(n)$, where $n$ is the number of sheets converging on edge $r$. Then $Z(h)=\{Z(h,r)\}$ is a set of laws of reflection and refraction in the state $X(h)$. Assume that $Z(h)$ is a piecewise constant function from $h$. For simplicity, let the function $Z(h)$ have only a finite number of jump discontinuities on the interval from $0$ to infinity, see Fig. 2. If the parameter $h$ varies within the regularity interval of the function $Z(h)$, then the laws of reflection and refractions on all edges of the state $X(h)$ remain unchanged. The law $Z(h)$ on a particular edge may change only when the parameter $h$ passes through the critical value (this is the jump discontinuity point of the piecewise constant function). 2. In some cases, for a critical $h$, an edge $r$ in the support complex $X$ becomes “penetrable” (“transparent billiard wall”). This means that, with growth of energy, a billiard ball now passes through it instead of being reflected inside the same sheet from which it arrived at the edge $r$. 3. We assume that in an integrable billiard the edge spines of the complex state $X(h)$ can change smoothly in the class of confocal quadrics when $h$ changes. Such an operation can be interpreted as billiard wall expansion. At critical values of the parameter $h$, they can merge with (be glued to) other edges (see Fig. 3), degenerate, and be transformed into segments of focal lines. That is, edge spines and free boundaries (impenetrable edges) in $X(h)$ are smooth functions from $h$. 4. We clarify the previous point. Let the regular value of $h$ change in an interval $D_i$ between two adjacent critical values (see Fig. 2), i.e., the topology of the state $X(h)$ has not changed yet. Assume that the boundaries of the glued sheets vary smoothly in the class of confocal quadrics. This condition is natural since in the integrable billiard theory this defines equivalent billiards [8]. Recall that billiard deformation in its class of equivalence does not change the topology of the Liouville foliation of its isoenergy three-dimensional surface (see [13]). In others words, transition to a dynamical system with Liouville equivalence takes place. At the same time, here (in a regular case) it is assumed that the boundary arcs of sheets do not fall on a focal line. We will provide a more detailed description of evolution of the complexes states $X(h)$ within the invariant support $X$. 2.2.2. Rearrangement of billiard states: spine gluing1. Operation related to gluing billiard sheets at a critical value of $h$. In case of this operation sheets are glued along boundaries, which are the same arc of one and the same quadric, see Fig. 3. For instance, we glue not only spines, but also free boundaries. Since several spines are glued to form one spine, then a new permutation cycle emerges on it. 2. When the jump occurs, we allow billiards bounded by confocal quadrics to change their equivalence class. Namely, let a boundary segment of a sheet at a critical value of $h$ fall on a focal line. In this case, before setting new permutations (see operations above), it is necessary to merge boundary segments of each flat sheet if a straight angle ($180$ degrees) was formed between them, or the segment “folded in half” (see example in Fig. 1). To achieve this, we need to make a cut on the sheet, but not force its banks apart. We will call it a new edge instead (keeping the permutation and the law of motion on it). Such a technical peculiarity is related to the fact that gluing rules for billiards have a number of constraints. For instance, gluing always takes place on segment boundaries (a segment is either a circle or an arc of a quadric extended from one right angle to another one). However, as we allow billiards to change their type at the moment of the jump, then the jump can make the angle of $90$ degrees equal to $180$ degrees. At this moment, the arc of the quadric ceases to be a segment, and it has to be integrated with other arcs, which we add (see example on Fig. 4). 3. Operation of integrating spines in boundary points. Spines of one state $X(h)$ may be glued at boundary points in the case when (as a result of the jump) they fell on one smooth boundary arc, i.e., when the angle between them became straight ($180$ degrees). For example, we consider a billiard bounded by two ellipses and two segments of the focal line. Let the smaller ellipse fall on the focal line in case of the jump. Then it is possible to integrate this smaller degenerated ellipse, namely the segment between the foci, with focal line segments to form a single segment, see Fig. 4. 4. All such successive transformations $\{Z(h,r)\}$, and spine gluing will be called jumps or rearrangements of the state $X(h)$ at critical values $h$. Consequently, starting from the initial state, we observe deformation jumps of subcomplexes $X(h)$ inside the invariant (“fixed”) evolutionary billiard support $X$. 2.2.3. Final definition of a force evolutionary integrable billiardDefinition 3 (Fomenko). a) The above-described complex $X$ is called the force evolutionary integrable billiard support. We believe that it is invariant and “immobile”. b) A family of expanding subcomplexes $X(h)$, “living inside” the support $X$, are called evolutionary billiard states, depending on $h$. Note that the support $X$ matches the last (“maximum”) state $X(N+\varepsilon)= X(\infty)$. We emphasize that the support $X$ is considered to be a topological complex, on the spines of which no permutations are indicated. c) Impenetrable edges of states $X(h)$ can become penetrable, but not vice versa. The edges can be glued. Boundary circles can shrink to points. d) An integrable system with two degrees of freedom defined by the billiard ball dynamics on the varying states $X(h)$ is called the force evolutionary integrable billiards (billiard system). e) The behavior of this system on regular and singular isoenergy three-dimensional surfaces (including “billiard ball sections” and the resulting multivalued flow) will be defined below. Remark 6. It may be difficult to seek for a connected evolutionary billiard, which naturally realizes a specific integrable Hamiltonian system (in the indicated sense). Seemingly, this issue could be solved in a trivial way. Consider successive isoenergy three-dimensional manifolds $Q_1,Q_2,\dots$, bearing the given integrable system on the relevant successive energy levels $h_1,h_2,\dots$ . We assume that the corresponding Liouville foliations on $Q_i$ are realized by billiard tables $A_i$. To produce an evolutionary billiard, the condition $X_i$ should be the billiard $A_i$. But if $A_i$ does not include $A_{i-1}$, then this cannot be done according to our rules. In other words, their “vulgar” gluing will immediately cause a new state $X$ to appear, and it will be different from states $X_i$ and $X_{i-1.}$ At some moment, in this state a billiard ball will start moving from one billiard to the neighboring one glued to it. Therefore, the system will start to “mix”. But in this case, the topology of the isoenergy three-dimensional manifolds and Liouville foliations on them change. Therefore, the original integrable system is replaced by another one. That is, such an attempt to realize the original system by billiards on all its isoenergy manifolds at once ends in failure. 2.3. Four-dimensional phase complexes of the evolutionary billiard table matching regular energy zonesLet $h$ – be a regular energy value from an interval $D_i$. The corresponding complex-state will be denoted as $X(D_i)$. Definition 4. The point of the phase complex $TX(D_i)$ is the pair $(x,v)$, where $x$ is a point on the billiard table $X(D_i)$, and $v$ a particle velocity vector in the point $x$. When the point $x$ is on the boundary of the sheet $L_i$, adjacent to the sheet $L_k$, the gluing of the corresponding pairs $(x,v)$ and $(x,w)$ ) occurs according to the law of reflection and refraction $Z(r, h)$, which applies on the given edge of the gluing $r$. Therefore, the evolutionary (force) billiard with support $X$ is defined with the data set $(X(D_i),\, \{X(h)\},\, \{Z(r,h),\, \text{gluings}\},\, N)$, where the integer $N$ defines partition of a real semiaxis of values $H=h$ from zero to infinity, defined by the numbers $0,1,2,3,\dots,N$. We would like to remind that the functions $X(h)$ and $Z(r,h)$ are constant on every open interval $(i,i+1)$. Points $1,2,\dots$ set the jumps of the reflection/refraction function, i.e., these are the critical values of the parameter $h$, see Fig. 2. The total number of intervals of constancy of the function $Z(r,h)$ is equal to $N+1$. Finite regularity intervals of this function will be denoted as $D_i$, i.e., $D_1,\dots,D_N$. The last one, which is already an infinite regularity interval, will be denoted as $D_\infty$. We would like to note that we do not consider zero to be a critical value because the material point velocity $v$ is always different from zero. 2.4. Regular isoenergy 3-surfaces of the evolutionary (force) billiardDefinition 5. A regular isoenergy 3-surface (complex) $Q_h$ is a subset in the four-dimensional phase complex $TX(D_i)$, defined by the equation $H=h$ (i.e., the “level of constant energy”), where $h$ is a regular parameter value. In the case of integrable billiard books, three-dimensional surfaces of constant energy matching regular values of $h$ are topologically three-dimensional manifolds (the Vedyushkina–Kharcheva theorem, see [74]). We will discuss singular surfaces $Q_h$, i.e., the surfaces matching critical values $h$, below. In general, this is not a manifold, but a cell complex with singularities. Formally, this definition matches the classical isoenergy surface concept for systems with two degrees of freedom, which are not necessarily integrable. These include two-dimensional topological billiards and billiard books. For a force billiard, each interval $D_i$, $D_\infty$ (where $i=1,2,\dots,N)$ generally matches its own regular isoenergy surface $Q_h$. Consequently, the number of such surfaces is $N+1$. Denote them as $Q_1,Q_2,\dots,Q_N,Q_\infty$ (see Fig. 2). Of course, it may turn out that some of them are homeomorphic to each other. Recall that for classical topological billiards and billiard books, the topology of the Liouville foliation on $Q^3_h$ does not depend on the choice of of a finite (non-zero) value $h$ of the energy $H=|v|^2$, i.e., all regular isoenergy 3-surfaces are of the same type. We will consider tables of integrable topological billiards and billiards books as important examples of billiard tables, with accuracy up to natural equivalences, see article by Vedyushkina [8]. 2.4.1. A visual commentThe idea of the evolutionary (force) billiard is new, as it takes into account the energy of a material point. Qualitative changes in a dynamical system with a change in the energy of a particle are studied, for example, in physics and quantum mechanics. When the energy increases, the electrons revolving around the nucleus of an atom “jump” from one energy level to another one. Therefore, “pumping” of energy causes bifurcations in a system. It turned out that something similar is also found in mathematical billiards. The walls of billiard tables become “sensitive” to the impact force of a material point. In other words, in the evolutionary billiard, the walls react to the energy of a point hitting the wall (each wall does it in its own way). With critical values of the energy, the walls change their properties, and the motion of a material point changes in accordance with the new law of reflection and refraction. 2.5. Singular isoenergy 3-surfaces of an evolutionary (force) billiardWe now consider how “singular” isoenergy 3-surfaces of an evolutionary (force) billiard are arranged. They are indicatively shown in Fig. 2 as three-dimensional surfaces $K_1,K_2,\dots,K_N$. They match singular energy values $h=1,2,3,\dots,N$. Let $h=i$ be the singular energy value. We denote the left billiard table state by $X(i-\varepsilon)$ and by $X(i+\varepsilon)$ the right billiard table state. Consider an edge spine $r$ on the left two-dimensional table $X(i-\varepsilon)$, on which the law of reflection will now change, as well as gluing will take place. The singular complex $X_i$ is arranged as follows. Take the complex $X(i-\varepsilon)$ ) and glue to the spine $r$ those sheets that should be glued to this spine after the given jump, i.e., in the complex $X(i+\varepsilon)$. Consider two “neighboring” regular three-dimensional manifolds: $Q_i$ (we call it the left one) and $Q_{i+1}$ (we will call it the right one), which match the tables $X(i-\varepsilon)$ and $X(i+\varepsilon)$. We define the singular 3-surface $K_i$, which is between them (see Fig. 2). We endow each point on the sheet of the complex $X_i$ with a velocity vector of length $h=i$. First, we identify, according to the standard law of reflection, the velocity vectors on those spines where the law of reflection does not change. Now we consider the spine on which the law of reflection changed. At each point of it we will identify all velocity vectors with the same direction if they were identified either in $Q_i$ or in $Q_{i+1}$. This leads to the fact that the three-dimensional surface $K_i$ we are constructing is compact. However, singularities that match these spines appear in it. This three-dimensional surface is generally not a three-dimensional manifold any more. The neighborhood of any point of the spine endowed with a velocity vector is no longer homeomorphic to a three-dimensional disk on the three-dimensional surface $K_i$. The surface $K_i$ is actually obtained from surface $Q_i$ by identifying the pairs $(x,v)$ (point-vector) with those pairs $(x,w)$ that must be identified in $Q_{i+1}$. In each case of multivaluedness, this will lead to identifying three point-vector pairs (one incoming to the spine and two outgoing ones) rather than two, as occurs on the spines in a regular case. It is this effect that leads to singularity. How is a singular three-dimensional surface arranged? It is not a topological manifold. It is a cell complex. It is a stratified three-dimensional manifold. Its strata are smooth manifolds. According to Fomenko’s conjecture, it is a semi-algebraic manifold. We described the topology of the singular three-dimensional surface $K_i$, “sandwiched” between two “neighboring” topological three-dimensional manifolds $Q_i$ and $Q_{i+1}$. Note that there is an analogy here with smooth integrated Hamiltonian systems with two degrees of freedom. In those systems singular isoenergy three-dimensional surfaces are also “sandwiched” between two adjacent regular three-dimensional surfaces of constant energy. For smooth systems, the singularity of a three-dimensional surface generally means that it is not a smooth manifold anymore, with $\operatorname{grad}(H)$ degenerating on it at some points. Moreover, the nature of degeneration can be quite diverse, and it depends, among other things, on the specific type of the Hamiltonian $H$. In the case of a force billiard, the picture is similar, and is described above. 2.6. Billiard flows on singular isoenergy three-dimensional surfaces of an evolutionary billiard. Splitting and partitioning of a billiard ball into two balls on singular two-dimensional billiard tablesNow we can figure out which “multivaluedness flow” is generated on the singular three-dimensional surface $K_i$ by “approaching” billiard flows on three-dimensional manifolds $Q_i$ and $Q_{i+1}$, when they “approximate” (on the left and on the right on $h$) to the three-dimensional surface $K_i$ “sandwiched” between them. Roughly speaking, each of these flows generates a flow on the singular three-dimensional surfaces $K_i$. These maximal flows are different. On the spines where the law of reflection changes after the jump, it is impossible to correctly determine the trajectory of the ball after reflection (refraction) (see Fig. 5). Note that we know how the ball behaves on the left $X(i-\varepsilon) $ and right $X(i+\varepsilon)$ billiard tables. When the ball reaches the spine given, before and after the jump it goes to different sheets (on the right and left tables). Informally, on the singular complex $X(i)$ the ball trajectory “bifurcates” after crossing this spine. That is, the ball moves on two sheets simultaneously, see Fig. 5. In other words, one can assume that after hitting such a spine the ball “splits” into two balls, and each of them “begins to live its own life”. So, when the energy value $h$ becomes equal to $i$, the billiard ball splits (divides) into two balls on the singular two-dimensional table. Each of the balls moves “on its own” sheet. In other words, approaching billiard flows on three-dimensional isoenergy surfaces “settle” in the limit on the singular three-dimensional surface $K_i$, generating a “branching flow” on it. Branching is induced by partitioning (splitting) of the ball into two ones at the moment of hitting the spine $r$. Note that this situation occurs only on those glued spines of the force billiard where the law of reflection changes in case of a jump. At the same time, an “elementary particle” splits into two ones. This is the difference from the smooth case. In the smooth case, on the singular isoenergy three-dimensional surface, a Hamiltonian flow with singularities determined by the singularities of the three-dimensional surface appears. This flow is single-valued in the sense that at each phase point of a three-dimensional surface one vector is “settled”. At the same time, for evolutionary billiards, the flow on a singular three-dimensional the surface is also singular. However, here it becomes branching. One set of its “branches” comes from the “right”, and the second set of “branches” comes from the “left”. This is why here in each singular phase point, two vectors are “settled”. § 3. Integrable billiards bounded by arcs of confocal quadricsWe fix the confocal quadric family with the ratio
$$
\begin{equation*}
(b-\lambda)x^2+(a-\lambda)y^2=(b-\lambda)(a-\lambda).
\end{equation*}
\notag
$$
Here $a$ and $b$ are fixed parameters of the family, which, for instance, fix the distance between the foci. If $a>b>0$, then this ratio describes a family of confocal ellipses and hyperbolas, which include the focal line $y=0$ and the limit hyperbola $x=0$. An elementary billiard means a compact connected part of a plane, the boundary of which consists of arcs of confocal quadrics and does not contain angles $3\pi/2$. We note that confocal quadrics always intersect at right angles. The prohibition of angles $3\pi/2$ allows us to correctly determine the billiard motion after a material point hits an angle. Namely, after reflection, the point continues its movement in the opposite direction along the same segment which has “led” it to the angle. Let the family of confocal quadrics consist of ellipses and hyperbolas. It makes sense to expand the set of elementary billiards by including coverings over the domain bounded by two ellipses, as well as parts of these coverings. On the set of elementary billiards, one can introduce the natural relation of equivalence, which preserves the Liouville foliation. Informally, two billiards are said to be equivalent, if one of them is obtained from another one by isometry of the plane or by changing the parameters of the boundaries to ensure that variable arcs of the boundaries do not change their type in case of deformation (for more details, see [8]). The definition forbids a segment of a variable boundary to change its type, i.e., in case of deformation segments remain either elliptical (i.e., the parameters of the quadrics on which these segments are located continuously vary within $(-\infty,b)$), or hyperbolic (i.e., parameters of the quadrics on which these segments are located continuously vary within $(b,a]$), or are always located on the focal line (all the time during deformation the parameter remains equal to $b$). At the same time, we repeat that in our assumption all ellipses and hyperbolas belong to the same family of confocal quadrics with parameters $a$ and $b$. § 4. The Euler caseWe consider specific examples of force evolutionary billiards realizing important integrable systems in geometry, mechanics, mathematics and physics. As the first example, we consider the famous Euler case in the dynamics of a heavy rigid body. We will show how the Euler case can be realized by a force evolutionary billiard on the entire phase manifold $M^4$ at a time, i.e., on all regular isoenergy three-dimensional surfaces. This system is described on the six-dimensional Lie algebra $e(3)$ of the motion group of the three-dimensional Euclidean space. In natural coordinates on the dual space $e(3)^*$ the Poisson bracket acquires the form
$$
\begin{equation*}
\{S_i, S_j\}=\varepsilon_{i j k} S_k, \qquad \{R_i,S_j\}=\varepsilon_{i j k} R_k, \qquad \{R_i, R_j\}=0,
\end{equation*}
\notag
$$
where $\{i, j, k\}=\{1,2,3\}$ and $\varepsilon_{i j k}=(i-j)(j-k)(k-i)/2$. The Hamiltonian system on $e(3)^*$ is described by the so-called Euler equations: where function $H$ is the Hamiltonian.We fix a symplectic leaf, i.e., a four-dimensional surface at the level of two integrals of the Euler equations: $f_1$ (the geometric integral) and $f_2$ (the area integral), and get
$$
\begin{equation*}
M_{c, g}^{4}=\{f_1=R_1^2+R_2^2+R_3^2=c,\, f_2=S_1 R_1+S_2 R_2+S_3 R_3=g\}.
\end{equation*}
\notag
$$
For almost all values of $c$ and $g$ g the joint surface at the level of functions is a smooth submanifold in $\mathrm{e}(3)^*$, on which the Poisson bracket is nondegenerate, which leads to the existence of a symplectic structure on this submanifold. Further we assume that $c$ and $g$ are regular values. The Euler case $(1750)$ describes the dynamics of a heavy rigid body fixed at the center of mass. The Hamiltonian and the additional integral have the form
$$
\begin{equation*}
H=\frac{S_1^2}{2 A_1}+\frac{S_2^2}{2 A_2}+\frac{S_3^2}{2 A_3}, \qquad K=S_1^2+S_2^2+S_3^2.
\end{equation*}
\notag
$$
We assume that $f_1=1$. Then different three-dimensional surfaces $Q^3$ are defined by the parameters $g$ and $h$. Consider the bifurcation diagram for the pair of the integrals $f_2$ and $H$. Curves of the bifurcation diagram partition the plane $R^2(g,h)$ in such a way that for all points $(g,h)$ from one domain the topological type of relevant isoenergy surfaces is the same. Consider the mapping
$$
\begin{equation*}
F=(f_2, H)\colon S^2\times \mathbb{R}^3\to \mathbb{R}^2(g,h),
\end{equation*}
\notag
$$
defined by the formula $F(P)=(f_2(P),H(P))\in \mathbb{R}^2(g,h)$. For the Euler case, the bifurcation diagram $(f_2,H)$ was calculated by Bolsinov and Fomenko (see [7], vol. II) and is shown in Fig. 6. For each of the resulting cameras, an integrable billiard had been previously found, which has Liouville equivalence to the Euler system (see [11]). Fig. 6 shows these five billiards, as well as the Fomenko–Zieschang invariants describing their Liouville foliations. At the same time, for each isoenergy surface a relevant billiard was discovered. As it was mentioned in the introduction, it has been discussed for a long time whether it is possible to realize the Hamiltonian system using billiards at once, entirely on $M^4$, simultaneously on all of its regular isoenergy surfaces. It turned out that evolutionary billiards allow this. We now show how this is achieved for the Euler case. 4.1. Construction of an evolutionary billiardWe fix the line $g=\mathrm{const}\neq0 $ on the bifurcation diagram (see Fig. 7). This line matches a symplectic leaf $M_g^4$, consisting of three pieces, and each of the pieces matches its isoenergy surface type. Denote critical values of $H$ by $h_0$, $h_1$ and $h_2$; at these values the isoenergy surface type changes. At $H\in(h_0,h_1)$ isoenergy surfaces are homeomorphic to the nonconnected union of two spheres $S^3$. At $H\in(h_1,h_2)$ the isoenergy surface is homeomorphic to the direct product $S^1\times S^2$, while at $H\in (h_2,\infty)$ the isoenergy surface is homeomorphic to the projective space $\mathbb{R}P^3$. Remark 7. In all subsequent figures of force evolutionary billiards, the arrows on billiard sheets show the trajectories of the billiard ball. Construct the force evolutionary billiard matching the given symplectic leaf $M^4_g$. It turns out that three of the four billiards shown in Fig. 6 are three states of the evolutionary billiard. The initial (starting) billiard is a nonconnected billiard that does not have common points with the focal line. It is homeomorphic to two disks. With the evolution the billiard is transformed into an annulus. Fig. 6 shows two annuli, and any of them is suitable for us. Lastly, at the final stage of the evolution the annulus is transformed into a sphere (ellipsoid). The resulting force billiard is shown in Fig. 1. We now provide a more detailed description of this process, indicating the billiard ball trajectory and gluing of spines. This more detailed evolution of the initial billiard is shown in Fig. 7 (in case of upward movement). We consider two disks glued along the boundary, bounded by the same ellipse. We obtain the surface $E$ homeomorphic to an ellipsoid. We fix the hyperbola $m$ with the parameter $\lambda_{m}>b$ and the ellipse $e$ with the parameter $\lambda_{e}<b$. Consider domains on the surface $E$ bounded by the ellipse $e$ and hyperbola $m$ fixed above. We select from these domains two domains that do not have common points with the focal line (see Fig. 7). Each of them is homeomorphic to a disk. The initial complex $X$ at $H\in(h_0,h_1)$ of the evolutionary billiard consists of this pair of domains. When the parameter $H$ increases, we change billiard boundaries, staying in the class of confocal quadrics. We direct the parameter $\lambda_{e}$ of the boundary ellipse $e$ to $b$ to make the parameter $\lambda_{e}$ have the value $b$ at $H=h_1$. At this moment the state of the billiard is modified. At the same time, we will glue the horizontal boundaries of two billiard tables to form an annulus (see Fig. 7). This annulus is a subset of the surface $E$ cut from it by two branches of the hyperbola $m$ with the parameter $\lambda_{m}$. At $H\,{\in}\,(h_1,h_2)$ we reduce the parameter $\lambda_{m}$ of the hyperbola $m$ to the value $b$. At $H=h_2$ we obtain the next jump. At the same time, in the force evolutionary billiard complex, boundary hyperbolas fell on the focal line. The angles between their arcs and segments of the focal line (along which there was gluing at the previous jump) became straight. This is why, when the jump occurs, first we cut the previously glued segments. Then we combine the segments that have common boundary points to form one segment. Finally, we glue all the billiard tables to form one table homeomorphic to the surface $E$ (see also Fig. 1). The constructed force evolutionary billiard includes two jumps (two transformations) between three variable billiards tables. Theorem 1. The evolutionary billiard constructed above, with the support homeomorphic to an ellipsoid, realizes (in the sense of Liouville equivalence) the integrable Euler case on the entire phase symplectic manifold $M^4_g$ once, i.e., on all its regular isoenergy three-dimensional surfaces for all regular values of both parameters $g$ and $h$. Note that the billiard walls evolve in the class of confocal quadrics, which ensures integrability of the system at every moment of its evolution. Proof. We use the theorem by Fomenko and Zieschang on the Liouville equivalence invariant. For this purpose, it is necessary to calculate the Fomenko–Zieschang invariants for all constructed billiards and check that they match the invariants encoding the Liouville foliations on isoenergy surfaces of the Euler case. To do this, it is necessary first to describe the bifurcations of Liouville tori, and calculate a coarse molecule — a Reeb graph, at the vertices of which bifurcation codes (the so-called atoms) are located. Then it is necessary to indicate how the bifurcations are glued to each other on boundary Liouville tori. To do this, according to the rules specified in [7], it is necessary to select admissible bases from the cycles $\lambda$, $\mu$ in the homology group of boundary tori and calculate transition matrices for transition from one admissible basis to another along the molecule edge connecting two selected bifurcations. Labels $r$, $\varepsilon$ (on the edges) and $n$, which are placed on the so-called “families”, need to be extracted from these gluing matrices. For the Euler case these invariants were calculated by Bolsinov and Fomenko [7], and for the above mentioned three types of billiards they were calculated by Vedyushkina [13]. Comparing these invariants (see Fig. 6), we find out that they are identical. This means, according to the Fomenko–Zieschang theorem, that these systems are Liouville equivalent. The theorem is proved. We consider a standard triaxial ellipsoid in $\mathbb{R}^3$. According to the Jacobi–Chasles theorem, tangents to the geodesic on the ellipsoid are tangents to a fixed hyperboloid confocal with the given ellipsoid. Consider a geodesic billiard in the domain cut on the ellipsoid by confocal one-sheeted and two-sheeted hyperboloids. A material point of the geodesic billiard moves in this domain along the segments of geodesics and is reflected from the boundaries according to the standard law. These billiards are integrable (see the book by Kozlov and Treshchev [53]), because the tangents to the point trajectory also touch some fixed hyperboloid (either one-sheeted or two-sheeted). Liouville foliations of such billiards with an accuracy up to Liouville equivalence were studied by Belozerov [51]. For instance, he calculated the Fomenko–Zieschang invariants for all such billiards and provided a complete classification of two-dimensional domains on an ellipsoid, which are integrable billiard tables. We select three billiards from them. The first billiard is represented by two domains homeomorphic to disks. These domains are simultaneously cut on the ellipsoid by confocal hyperboloids (a one-sheeted and a two-sheeted ones). At the same time, these two disks intersect the middle semiaxis of the ellipsoid (see Fig. 1). The second billiard on the ellipsoid is bounded by a one-sheeted hyperboloid. The third one is represented by the whole ellipsoid. As it was shown by Belozerov [51], each of these billiards is Liouville equivalent to the Euler case on the relevant isoenergy surface. Namely, the first billiard realizes the Liouville foliation for the Euler case on the nonconnected union of two three-dimensional spheres $S^3$. The second one does so on the direct product $S^1\times S^2$, while the third one acts on the projective space $\mathbb{R}P^3$. We construct one evolutionary billiard based on these three billiards (see Fig. 1). Let the first billiard described above and homeomorphic to two disks be the initial billiard. During evolution, two sheets of the first billiard expand, gradually filling the annulus cut out by the one-sheet hyperboloid. At the moment of the jump, they are glued to form the annulus which is the second billiard. Then this annulus continues to expand on the ellipsoid and gradually fills it completely. At the moment of the last jump it is transformed into a complete two-dimensional ellipsoid (see Fig. 1). We repeat that in case of this evolution the boundaries (walls) of the billiard are deformed in the class of confocal quadrics. Theorem 2. The constructed geodesic evolutionary billiard on the two-dimensional ellipsoid realizes (in the sense of Liouville equivalence) the integrable Euler case on the entire phase symplectic manifold $M^4_g $ at once, i.e., on all its regular isoenergy three-dimensional surfaces for all regular values of both parameters $g$ and $h$. Note that the billiard walls evolve in the class of arcs cut by confocal hyperboloids on the ellipsoid. This provides integrability of the system at every moment of its evolution. The ellipsoid is the support of this evolutionary billiard. Proof. As it was stated above, according to the theorem by Belozerov [51] when no jumps occur, each billiard table state fiberwise models a system on an isoenergy surface of the Euler case. This fact was also proved by comparing the relevant Fomenko–Zieschang invariants. In case of the billiard evolution described above, the isoenergy surfaces change their types in the same order in which the isoenergy surface $Q^3$ in the symplectic leaf $M^4_g$ of the Euler case changes its type. The theorem is proved. Remark 8. We remind that, according to the theorem by Bolsinov and Fomenko, the Euler case is Liouville (and even continuously trajectory) equivalent to the Jacobi problem, i.e., a geodesic flow on a two-dimensional ellipsoid (see [7]). In the case of the force evolutionary billiard, the triaxial ellipsoid emerges again. Remark 9. The force evolutionary billiard constructed by us is actually realized on one two-dimensional surface: the support homeomorphic to the ellipsoid. In other words, changes and transformations of the indicated states of the evolutionary billiard “live” on the same ellipsoid which is the support (see Figs. 1 and 7). § 5. The Lagrange caseThe Lagrange case describes the motion of an axisymmetric heavy rigid body with a point fixed on the axis of symmetry. The integrals have the following form (here $A$ and $B$ are parameters of the top):
$$
\begin{equation*}
H=\frac{S_1^2}{2 A}+\frac{S_2^2}{2 A}+\frac{S_3^2}{2 B}+a R_3, \qquad K=S_3.
\end{equation*}
\notag
$$
Bolsinov and Fomenko showed what depending on the parameters of the system there are four types of bifurcation diagrams [7]. All of them are shown in Fig. 8. Note that despite the presence in each case of the bifurcation diagram of its own sets of regular symplectic leaves, overall, in the Lagrange case, there are exactly five different types of symplectic leaves. 5.1. Topological billiards matching cameras of bifurcation diagrams in the Lagrange caseAll such billiards are bounded with arcs of concentric circles. $\bullet$ To realize the Lagrange system on the three-dimensional sphere $S^3$, we take the topological billiard glued from a disk bounded by a circle, and an annulus bounded by the same circle and the circle with a smaller radius. $\bullet$ To realize the Lagrange system on the direct product $S^1\times S^2$, we consider two annuli glued along the common circle with a larger radius. $\bullet$ To realize the system on the projective space $\mathbb{R}P^3$, we consider two discs glued to each other. Fig. 9 shows one of these disks divided into a smaller disk and an annulus. $\bullet$ To realize the system on a nonconnected sum $S^3 \cup(S^1\times S^2)$, we consider a billiard constructed for realizing $S^3$, and the annulus in which the radius of its larger boundary circle does not exceed the radius of the smaller circle of the annulus taken to realize $S^3$ (see Fig. 9). $\bullet$ To implement the Lagrange system on a disconnected $2S^3$, we consider the billiard constructed taken to realize $S^3$, and the disk with the radius of the boundary circle not exceeding the radius of the smaller circle of the annulus taken to realize another instance of $S^3$ (see Fig. 9). 5.2. Classification of transformations of topological billiard states during transitions between cameras of bifurcation diagramsProposition 1. Evolutions of Hamiltonian systems in the Lagrange case for transitions through arcs of bifurcation diagrams can be realized by the following seven transformations of relevant billiard states (see Fig. 10). 1. In case of the transformation $S^3\to \mathbb{R}P^3$, the inner circle of the ring glued to the disk shrinks to a point. The result is a billiard table homeomorphic to the sphere and glued from two disks. 2. In case of the transformation $S^1\times S^2\to S^3$, the inner circle of one of the annuli shrinks to a point. A disk is obtained. 3. In case of the transformation $S^3\to 2S^3$, a small disk is generated from the selected point, whose boundary circle radius does not exceed the smaller radius of the annulus. A two-dimensional surface becomes homeomorphic to a nonconnected union of two disks. 4. In case of the transformation $2S^3\to \mathbb{R}P^3$, the boundary circle of the disk is glued to the boundary circle of the annulus glued to the larger disk. The obtained billiard is obviously represented by two disks glued on the boundary, i.e., it is homeomorphic to the sphere. 5. In case of the transformation $S^3\to S^3\cup(S^1\times S^2)$, a new instance $S^1\times S^2$ is obtained from the circle, swelling up to form an annulus with radii of boundary circles not exceeding the radius of the smaller circle of the annulus for realizing $S^3$. 6. In case of the transformation $S^3\cup(S^1\times S^2)\to S^3$, the larger circle of the annulus is glued to the smaller boundary circle of the topological billiard that matches $S^3$. The obtained billiard is homeomorphic to the disk. 7. In case of the transformation $S^3\cup(S^1\times S^2)\to 2S^3$, the larger circle of the annulus matching $S^1\times S^2$, shrinks to a point. This results in a nonconnected union of two disks. 5.3. Force evolutionary billiards for the Lagrange caseWhen analyzing bifurcation diagrams for the Lagrange case (see Fig. 8), we find out that all symplectic leaves belong to one of the five types. The Hamiltonian system on each symplectic leaf is defined with increasing energy $h$ by a chain of transformations of its invariants, i.e., labeled molecules. In this case, a chain of transformations of the relevant isoenergy three-dimensional surfaces is also modelled. We construct an appropriate evolutionary billiard for each type of symplectic leaves. 1. The symplectic leaf of the first type determines the following chain of transformations: $S^1\times S^2 \to S^3\to \mathbb{R}P^3 $ (such a symplectic leaf appears in the bifurcation diagram for the case (a), see Fig. 8). The initial billiard state is represented by two annuli bounded by two concentric circles and glued along the circle with the larger radius. In case of the transformation $S^1\times S^2 \to S^3$, the inner circle of one of the annuli shrinks to a point. In case of the transformation $S^3\to \mathbb{R}P^3$, the internal circle of the other annulus shrinks to a point. The billiard state becomes homeomorphic to the sphere. The relevant evolutionary billiards will be denoted by $\operatorname{Bill}_1$ (see Fig. 11). 2. The symplectic leaf of the second type determines the transformation $S^3\to \mathbb{R}P^3 $ (such a symplectic leaf appears in all four types of bifurcation diagrams and matches, for example, $g=0$). The initial billiard state is an annulus bounded by two concentric circles and glued along the circle with the larger radius to the disk. In case of the transformation $S^3\to \mathbb{R}P^3 $ the inner circle of this annulus shrinks to a point. The billiard state becomes homeomorphic to the sphere. The relevant evolutionary billiard will be denoted as $\operatorname{Bill}_2$. 3. The symplectic leaf of the third type determines the following chain of transformations: $S^3\,{\to}\, 2S^3\,{\to}\, \mathbb{R}P^3 $ (such a symplectic leaf appears in the bifurcation diagrams (c) and (d), see Fig. 8). The initial billiard state is an annulus bounded by two concentric circles and glued along the circle with the larger radius to the disk. In case of the transformation $S^3 \to 2S^3$, a new disk is generated from a point, bounded by a circle with a small radius. Then the radius of this circle increases until it matches the radius of the smaller circle of the original annulus. At the moment of transformation $2S^3\to \mathbb{R}P^3 $ two billiards are glued along the boundary circles (a sphere is formed). The relevant evolutionary billiard will be denoted by $\operatorname{Bill}_3$. 4. The symplectic leaf of the fourth type determines the following chain of transformations: $S^3 \to S^3\cup (S^1\times S^2)\to 2S^3\to \mathbb{R}P^3 $ (such a symplectic leaf appears only in the bifurcation diagram (d), see Fig. 8). The initial billiard state is an annulus bounded by two concentric circles and glued along the circle with the larger radius to the disk. In case of the transformation $S^3 \to S^3\cup (S^1\times S^2)$, a new annulus is generated from the circle with the small radius (which is smaller than the radius of the inner circle of the original annulus). This new annulus is bounded by circles with the small radius. Then the inner circle of this annulus shrinks to a point, which matches the transformation $ S^3\cup (S^1\times S^2)\to 2S^3$. Further, the boundary circle of the formed disk increases until it matches the boundary circle of the original ring. In case of the transformation $ 2S^3\to \mathbb{R}P^3 $, gluing of two disks occurs (recall that in the initial billiard the disk was already glued to the initial annulus). Fig. 12 shows the sphere which is homeomorphic to the support of the given evolutionary billiard, and its states that are homeomorphic to the changing domains on the sphere. 5. The symplectic leaf of the fifth type defines the following chain of transformations: $S^3 \to S^3\cup (S^1\times S^2)\to S^3\to RP^3$ (such a symplectic leaf appears only in the bifurcation diagram (d), see Fig. 8). The initial billiard state is the annulus bounded by two concentric circles and glued along the circle with the larger radius to the disk. In case of the transformation $S^3 \to S^3\cup (S^1\times S^2)$, a new annulus is generated from the circle with the small radius (which is smaller than the radius of the inner circle of the original annulus). This new annulus is bounded by circles with the small radius. Further, the large circle of this annulus increases, and at the moment of the transformation $ S^3\cup (S^1\times S^2)\to S^3$ a gluing of a new annulus with the initial billiard occurs. The transformation $ S^3\to RP^3$ results from the contraction of the boundary circle to a point. This evolutionary billiard will be denoted by $\operatorname{Bill}_5$. Fig. 13 shows billiard states as domains on the two-dimensional sphere. Theorem 3. The integrable Lagrange case is realized (in the sense of Liouville equivalence) on each of its regular symplectic four-dimensional leaf $M^4_g$ by one of the force evolutionary billiards $\operatorname{Bill}_1$–$\operatorname{Bill}_5$ described above. Their billiard states are bounded by concentric circles. Note that the evolution of billiard walls takes place in the class of concentric circles, which provides the integrability of the system at every moment of its evolution. Proof. The Fomenko–Zieschang invariants for systems on the regular isoenergy surfaces of billiards $\operatorname{Bill}_1$–$\operatorname{Bill}_5 $ were calculated in the work [34]. The obtained invariants have the form $A$ – $A$. The label $r$ is equal to zero if the billiard is homeomorphic to the disk. It is equal to $1/2$ if the billiard is homeomorphic to the sphere, and is infinite in the case when the billiard is homeomorphic to the annulus. The label $\varepsilon$ is equal to $1$. Comparing the obtained invariants with the invariants calculated for the Lagrange cases [7], we find out that they coincide. This provides Liouville equivalence of the systems under consideration. The theorem is proved.
§ 6. Billiard transformation of the Euler case into the Lagrange caseIn this section, we will present the fact we discovered, demonstrating the value of evolutionary billiards in modeling integrable systems. We consider two well-known cases of integrability in the dynamics of a heavy rigid body. These are the Euler case and the Lagrange case. These cases differ qualitatively. In particular, the integral of the Euler case is quadratic, and the integral of the Lagrange case is linear. This explains the significant difference in the topology of these systems. Billiards made it possible to discover a unique fact which cannot be revealed using the classical approach to these systems. The Euler case “lives” on one regular symplectic four-dimensional manifold. As we showed, it is realized by the force evolutionary billiard described above. The Lagrange case “lives” on five regular four-dimensional manifolds (which match different values of the integral of areas $g$). As we showed, on each of these symplectic four-dimensional leaves, the Lagrange case is realized by the relevant force billiard. We revealed that there is an interesting (“hidden”) connection between the “Euler force billiard” and the five “Lagrange force billiards”. Consider an evolutionary billiard realizing the Euler case. Each of the topological billiards is bounded by arcs of confocal ellipses and hyperbolas. If we direct the foci towards each other, the ellipses will be transformed into concentric circles, and each hyperbola will be transformed into a pair of straight lines crossing the center of the above-mentioned circles (i.e., to their asymptotes). It turns out that the Liouville foliations of regular isoenergy surfaces of the Euler case will be transformed into the Liouville foliations of all three types of regular isoenergy surfaces of the Lagrange case. Theorem 4. Consider an evolutionary (force) billiard which simulates the Euler case. By directing focuses towards each other, we deform the boundaries of this billiard: the family of confocal ellipses and hyperbolas is transformed into a family of concentric circles and straight lines passing through a common center. Then this billiard (for the Euler case) will be transformed into a new evolutionary billiard, in which the full set of Liouville foliations matches the full set of Liouville foliations of the Lagrange case. We will call such systems “billiard equivalent”. Remark 10. Note that this discovered “transformation” of the Euler case into the Lagrange case does not transform the symplectic leaf of the Euler case into any of the five types of symplectic leaves of the Lagrange case. It is more complex. A full set of Liouville foliations of the Euler case is transformed into a complete set of Liouville foliations of the Lagrange case. At the same time, the order and even the number of connected components of regular isoenergy surfaces change. For instance, it was precisely this circumstance that, in the classical approach, did not allow us to notice the transformation of these systems into each other earlier. It was revealed that for such a transformation, first it was necessary to discover “hidden confocal quadrics” in the Euler case, and in the Lagrange case “hidden concentric circles” were to be revealed. As a result, it is natural and simple deformation of confocal quadrics in a circle (when foci merge) that “transforms” the Euler case into the Lagrange case (in the sense of Liouville equivalence). Here we actually introduced a new operation on systems that allow billiard realization. Let one system $V$ be realized by a billiard on a family of confocal ellipses (hyperbolas), and let another system $W$ be realized by a billiard on a family of concentric circles. We assume that when the billiard is deformed, namely, when the foci of the ellipses merge into one point, the first system is transformed into the second one (in the sense specified above). Definition 6. We will state that such integrable Hamiltonian systems $V$ and $W$ are “billiard equivalent”. Proof of theorem 4. Fig. 14 shows how the Fomenko–Zieschang invariants change when such a transformation of the force billiard occurs. Almost all of them can be extracted from [8], [34]. We only need to show that a billiard homeomorphic to a disk retains its invariant when transformation of a family of confocal ellipses and hyperbolas into concentric circles and straight lines passing through their centers takes place.
Let the boundary of a flat billiard consist of arcs of concentric circles, with the center at the origin and, perhaps, straight lines passing through the origin. Then any billiard trajectory touches a circle (its radius may even be equal to zero), with a center at the origin. The radius of this circle or a directed angle between the trajectory and the fixed circle, for example, the billiard boundary, can be used as an additional integral. We prove that the Liouville foliation of the isoenergy surface for the billiard depicted in Fig. 14, top right, matches the invariant $A$ – $A$, labeled $r=0$, $\varepsilon=1$. Here, it is convenient to use the radius of the circle touched by the trajectory as an additional integral. Then one circle obviously corresponds to the value of the integral matching the radius of the large circle (a convex arc of the billiard gluing). This circle is a trajectory that runs along the convex arc of the billiard. Trajectories lying on the lines which pass through the origin correspond to the zero value of the integral. Such trajectories form an isointegral surface homeomorphic to the annulus which is the direct product of the circle and the segment. Here the circle is an arbitrary trajectory, and the segment is an arc of a concentric circle located inside the domain and equipped with velocity vectors (inward or outward). It is easy to understand that all remaining isointegral surfaces are homeomorphic to two-dimensional tori. Therefore, the isoenergy surface is glued together from two solid tori. It is obvious that the axis of each solid torus shrinks to a point inside the other one (see Fig. 15), which means that the label $r$ in the molecule $A$ – $A$ is equal to zero (while the label $\varepsilon$ depends on the orientation and may be assumed to be equal to one). Theorem 4 is proved. Remark 11. Type $A $ bifurcations that arise in smooth Hamiltonian systems describe the contraction of tori to a circle. In our case, contraction of tori to the annulus is a bifurcation of such type. Nevertheless, in both types of the obtained solid tori, the cycle $\lambda$ shrinking to a point inside the solid torus and the cycle $\mu$ that is homotopic to arbitrary integral trajectories on the annulus are defined correctly. § 7. Force billiards and the Goryachev–Chaplygin–Sretensky caseThe Goryachev–Chaplygin case describes the motion of a heavy rigid body with a fixed point and special symmetry conditions indicated below. At the same time, the integral of energy $H$ and the additional integral $K$ have the following form:
$$
\begin{equation*}
\begin{aligned} \, H &=\frac{S_1^2}{2 A}+\frac{S_2^2}{2 A}+\frac{2 S_3^2}{A}+a_1 R_1+a_2 R_2, \\ K&=S_3(S_1^2+S_2^2)-A R_3(a_1 S_1+a_2 S_2). \end{aligned}
\end{equation*}
\notag
$$
Here the integral $K$ is of the third degree. In this case, the center of mass of the body is located in the plane of the body symmetry matching the first two axes of the body inertia, i.e., in the equatorial plane of the inertia ellipsoid. Here the Poisson bracket for functions $H$ and $K$ has the following form: Therefore, it is obvious that the functions $H$ and $K$ are not involutions on all four-dimensional manifolds $M^4_{1,g}$. This is why the system is integrable on only one special 4-surface $\{f_1=1,\ f_2=0\}$, i.e., on $M^4_{1,0}$. This is the case of the so-called partial integrability corresponding to the zero value of the area integral $f_2$. The Sretensky case describes the motion of a gyrostat in the gravity field. The pair of integrals $H$ and $K$ has the following form:
$$
\begin{equation*}
\begin{aligned} \, H &=\frac{S_1^2}{2 A}+\frac{S_2^2}{2 A}+\frac{2(S_3+\lambda)^2}{A}+a_1 R_1+a_2 R_2, \\ K &=(S_3+2 \lambda)(S_1^2+S_2^2)-A R_3(a_1 S_1+a_2 S_2). \end{aligned}
\end{equation*}
\notag
$$
This case is a generalization of the Goryachev–Chaplygin case, which results from it when the parameter $\lambda$ is equal to zero. Here, like in the Goryachev–Chaplygin case, the system is integrable only on one four-dimensional surface $\{f_1=1,\ f_2=0\}$. The additional integral exists only on one four-dimensional surface $\{f_1=1$, $f_2= 0\}$. Therefore, to describe the invariants of this integrable case, there is no need to examine the mapping of the moment $(f_2, H)$. The topological type $Q^3$ and the Fomenko invariant for $Q^3$ 3 in this case depend on the value of the parameter $\lambda$ in the system and the value $h$, defining the isoenergy three-dimensional surface $Q^3_h=\{f_1=1,\, f_2=0,\, H=h\}$. This is why for the Sretensky case, the bifurcation diagram is constructed on the plane $\mathbb{R}^2(\lambda,h)$. We clarify that the arcs of the bifurcation diagram separate different types of isoenergy surfaces. On the same plane $\mathbb{R}^2(\lambda,h)$, the curves separating homeomorphic isoenergy surfaces which have different Liouville foliations are also depicted. Fig. 16 shows the bifurcation diagram with separating curves. For some obtained cameras, an integrable billiard which is Liouville equivalent to the Sretensky system was found. These cameras have a darker color in Fig. 16. We single out two symplectic leaves $A$ and $B$ located in the preimages of dashed lines (also denoted by $A$ and $B$) in Fig. 16. Lines $A$ and $B$ pass through the dark domains on the bifurcation diagram, i.e., we can find an evolutionary billiard which partially models the Liouville foliations on the corresponding symplectic leaves. 7.1. Detailed description of force evolutionary billiards $A$ and $B$The ambient support of the evolutionary billiard $A$ (see Fig. 17 on the top on the right) consists of the three billiards. This is the billiard $A_1$ containing one focus and bounded by an arc of an ellipse and an arc of a hyperbola and two congruent quadrilateral billiards of class $B_1$, bounded by the arcs of the same ellipse and hyperbola as the billiard $A_1$, and the arc of the larger ellipse. The initial state is the billiard consisting of three pieces: one billiard $A_1'$ (a half of the billiard $A_1$) and two billiards $B_1'$(halves of the billiard $B_1$). One of the billiards $B_1'$ is glued to the billiard $A_1'$ along the arc of the smaller ellipse (in Fig. 17 this is the gluing along the dotted line), and along the arc of the larger ellipse to the other billiard $B_1'$. At the moment of the first jump, gluing occurs along the concave elliptical boundaries of the billiards $B_1'$. At the moment of the next jump, each of the constituent billiards expands, gluing to an equal billiard on the other side of the focal line (i.e., we can say that in the initial complex the focal line becomes a penetrable wall). The evolutionary billiard $B$ (see Fig. 17) describes one jump which is the transition from half of the billiard $A_1 $ to the billiard $A_1$. With such a jump, as in the previous case, the billiard wall lying on the focal line becomes penetrable. Theorem 5. The constructed evolutionary billiards $A$ and $B$ (see Fig. 17) realize (in the sense of Liouville equivalence) the integrable Goryachev–Chaplygin–Sretensky case on part of the phase symplectic manifolds $M^4_\lambda$, that match the lines $A $ and $B$ in Fig. 16. We emphasize that the billiard walls evolve in the class of confocal quadrics, which provides integrability of the system at each moment of its evolution. The proof of the theorem follows from the calculated Fomenko–Zieschang invariants, which can be found in [7] and [45]. The authors thank V. A. Kibkalo for a number of valuable remarks, as well as the peer reviewer for attention to the work and many comments that contributed to the improvement of the text. 
Citation in format AMSBIB
\Bibitem{FomVed22} |
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