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This article is cited in 1 scientific paper (total in 1 paper) Topological analysis of pseudo-Euclidean Euler top for special values of the parameters M. K. Altuev, V. A. Kibkalo Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 3, DOI: https://doi.org/10.4213/sm9771 
Bibliographic databases:
    § 1. IntroductionInvestigations of Hamiltonian systems (and, in particular, the properties of their special motions, namely, periodic trajectories and equilibria) by means of topological methods and Morse theory approaches was significantly prompted by Smale’s well-known paper [1]. Consider a level of energy $H$ in the phase space of the Hamiltonian system $v=\operatorname{sgrad}H$ (fixing also some values of the parameters of the system; for example, in many mechanical systems the values of the geometric and area integrals are such parameters). Whether or not there are special motions in the system on this level is often connected with the presence of critical points of $H$ on the corresponding isoenergy surface $Q^3_h$, which is invariant under the Hamiltonian vector field $\operatorname{sgrad}H$. It turns out that in many cases going over bifurcation values of the energy entails the change of the topological type, that is, homeomorphism class, of these 3-manifolds. If the Hamiltonian system is Liouville integrable (see [2] for details), then its phase space is partitioned into common levels of first integrals, that is, it has the structure of a Liouville foliation. By Liouville’s theorem, provided that the system is nonresonant, almost all leaves in this foliation are the closures of phase trajectories. In other words a leafwise homeomorphism between two systems means that almost all closures of trajectories of one system can simultaneously be mapped to the closures of trajectories of the other system. An analogue of Morse’s theory for integrable Hamiltonian system was developed by Fomenko [3], [4]. Subsequently, under the assumption that an additional first integral $F$ of the system is of Bott type on a nonsingular ($d H \ne 0$) isoenergy surface $Q^3_h$ of this system and a number of other conditions are met, Fomenko and his school developed a theory of topological classification of Liouville integrable Hamiltonian systems; see [5], [6] and also [2] for details. One can compare two systems from the standpoint of Liouville equivalence (up to a leafwise homeomorphism of the restrictions of these systems to $Q^3$) or rough Liouville equivalence (up to a homeomorphism of the bases of the corresponding Liouville foliations on $Q^3$ which lifts to a leafwise homeomorphism of the foliations in a neighbourhood of each point in the base) by comparing the relevant classifying invariants. For rough Liouville equivalence this is the Fomenko invariant (rough molecule), while for Liouville equivalence this is the Fomenko-Zieschang invariant (molecule with numerical marks). These invariants are the Reeb graphs of the restriction of the function $F$ to the isoenergy surface $Q^3$ of the system which are endowed with certain additional data. Recall that edges of such a graph correspond to families of regular Liouville tori and vertices correspond to singular leaves (when $Q^3$ is compact, the preimage of a bifurcation value of $F$ contains singular points). Classes of such singularities with respect to leafwise homeomorphisms are called atoms. Numerical marks on a Fomenko-Zieschang invariant are determined by the gluing homeomorphisms of the boundary tori of two atoms sharing the same edge. Since our paper is devoted to a considerable extent to finding types of bifurcations (not necessarily possessing critical points in the preimage of a bifurcation value of the moment map) and analogues of Fomenko invariants for concrete systems with noncompact Liouville foliations we recall more facts on these invariants in the compact case. Each Bott 3-atom of the system contains at least one circle of critical points (namely, this is the singular leaf of the atom). We assume below that these points are nondegenerate. If this circle is minimal or maximal (of elliptic type), then the singular leaf coincides with this circle, and an invariant neighbourhood of it (the 3-atom) can be represented as an $S^1$-bundle over a disc foliated by circles and a Morse singular point of centre type. Such a 3-atom (as well as its base, the disc foliated as above) is denoted by $A$. If some critical circle of a 3-atom has the saddle type, then all other critical points on this singular level (if there are any) are saddle points too, and they form saddle circles. In this case the singular level contains both critical and regular points. The 3-atom itself has the structure of a Seifert $S^1$-fibration, with singular fibres of type $(1,2)$; see [2] and [7] for details. In other words, each Bott 3-atom has the structure of an $S^1$-fibration over a compact 2-dimensional base. This base is a neighbourhood of a critical level of a Hamiltonian system with one degree of freedom and Morse critical points. In Figure 1 we show foliated neighbourhoods of critical points of centre and saddle type (we denote the latter by $B''$ as in [8]–[10]) and present examples of saddle 2-atoms: an atom $B$ (a foliation in an invariant neighbourhood of a ‘figure-of-eight’) and an atom of type $C_2$, which we also encounter in what follows. The notation $B''$ is used for a More saddle because we can obtain an atom $B$ from this saddle by gluing it along two pairs of boundary segments. 1.1. Foliations with noncompact leaves and incomplete flowsThe condition of Liouville integrability imposes significant restrictions on the system, alongside the involutivity of the first integrals (the property of commuting with respect to the symplectic structure or Poisson bracket). The first of these restriction is that the Hamiltonian flows of the first integrals must be complete, so that we can shift along trajectories of these fields by any value of time $t \in \mathbb{R}$. In other words, the system cannot go far away without limit in finite time along a phase trajectory. If this fails for a first integral of the system, then we say that the corresponding Hamiltonian vector field is incomplete. Investigations of systems with incomplete fields can often be complicated, and showing that a flow is complete or not can be complicated too. Nonetheless, these questions can be answered for some systems: see [11]–[13]. We also mention several more general results concerning generalizations of Liouville’s theorem to systems with incomplete flows: see [14] and [15]. A rather convenient condition ensuring the completeness of Hamiltonian vector fields on a leaf of the Liouville foliation is the compactness of this leaf. For systems with polynomial integrals we need not work with a system of differential equations to verify this condition: this often reduces to looking at a common level of several functions from the points of view of its boundedness and separation from certain points. Note that on noncompact leaves the system can have complete and incomplete flows of the Hamiltonian fields of first integrals alike. One of the main problems in the topological analysis of systems with noncompact leaves is the presence of noncritical bifurcations of Liouville foliations. In other words, the preimage of a neighbourhood of some value of the moment map of such a system may not be a trivial fibration over this neighbourhood but the preimage of the value itself may contain no critical points (where the rank of the differential of the moment map drops). In particular, we cannot find such singularities by verifying whether the Hamiltonian fields for first integrals are linearly dependent; see [12] and [13]. Recently an overview of the phenomenon of noncompactness in integrable systems was made by Fedoseev and Fomenko [16]. They presented a list of examples of mechanical systems of different nature such that their Liouville foliations contain noncompact leaves and bifurcations of these leaves. In particular, note recent papers on the classification and topological analysis of various generalizations of Bertrand’s classical systems, for instance, [17] and [18]. Singularities with noncompact foliations were also the subject of Nikolaenko’s recent papers [9] and [19]. One example of an explicit solution of the problem of whether a common level of first integrals of a mechanical system is compact is Kibkalo’s recent paper [20], where a compactness criterion was established for a pseudo-Euclidean analogue of the well-known Kovalevskaya top. He was also able to show that the Liouville foliation for this system has noncompact noncritical bifurcations, although their precise form it is not yet clear. Previously, in [21] a separation of variables was proposed for this system. A class of pseudo-Euclidean analogues of integrable systems of classical dynamics (the Euler, Lagrange and Kovalevskaya tops, the Goryachev-Chaplygin and Hess systems) was introduced by Borisov and Mamaev [22]. We can also mention [23], a collection of classical and recent papers devoted to problems of mechanics in non-Euclidean spaces. One reason why we turn to the topology of the pseudo-Euclidean Euler top is the simpler form of the Hamiltonian and additional first integral in comparison with the Kovalevskaya top. It could be anticipated that the bifurcations of foliations arising there can be described explicitly and geometrically. In Figure 2 we present examples of bases of noncompact bifurcation atoms in which there are no critical points (and maybe nor even singular leaves) that occur in pseudo-Euclidean analogues of the Euler system for various values of the parameters of this system. 1.2. Pseudo-Euclidean analogues of systems of dynamicsConsider a one-parameter pencil $P$ of Lie-Poisson brackets on $\mathbb{R}^6(J_1, J_2, J_3, x_1, x_2, x_3)$, which for $\varkappa <0$, $\varkappa=0$ and $\varkappa >0$ correspond to the Lie algebras $\mathrm{so}(3,1)$, $\mathrm{e}(3)$ and $\mathrm{so}(4)$, respectively (below $\varepsilon_{ijk}$ is the sign of the permutation $123 \to ijk$):
$$
\begin{equation*}
\{J_i, J_j\}=\varepsilon_{ijk} J_k, \qquad \{J_i, x_j\}=\varepsilon_{ijk} x_k, \qquad \{x_i, x_j\}=\varkappa \varepsilon_{ijk} J_k.
\end{equation*}
\notag
$$
Let $g$ denote the pseudo-Euclidean scalar product $g=\mathrm{diag}(1, 1, \sigma)$ in some space $\mathbb{R}^3$, where $\sigma < 0$. Here and below we consider the case when $\sigma=-1$ (the value of this parameter is not essential). Now consider the transformation $\Phi\colon \mathbb{C}^3 \to \mathbb{C}^3$ of the form
$$
\begin{equation*}
\widetilde{J_j}=\frac{J_j}{i}, \quad \widetilde{x_j}=\frac{x_j}{i} \quad \text{for } j=1, 2, \qquad \widetilde{J_3}=J_3, \quad \widetilde{x_3}=x_3.
\end{equation*}
\notag
$$
It takes Poisson brackets in $P$ to brackets in a pencil $P_{\mathrm{ps}}=P_{\mathrm{ps}}(\varkappa)$ so that the Lie algebra $\mathrm{e}(3)$ of the group of motions of three-dimensional Euclidean space (corresponding to $\varkappa=0$) is taken to the Lie algebra $\mathrm{e}(2,1)$ of the group of motions of a pseudo-Euclidean space:
$$
\begin{equation*}
P_{\mathrm{ps}}=\begin{pmatrix} 0 & \sigma \widetilde{J}_3 & -\widetilde{J}_2 & 0 & \sigma \widetilde{x}_3 & -\widetilde{x}_2 \\ -\sigma \widetilde{J}_3 & 0 & \widetilde{J}_1 & -\sigma \widetilde{x}_3 & 0 & \widetilde{x}_1 \\ \widetilde{J}_2 & -\widetilde{J}_1 & 0 & \widetilde{x}_2 & -\widetilde{x}_1 & 0 \\ 0 & \sigma \widetilde{x}_3 & -\widetilde{x}_2 & 0 & \varkappa \sigma \widetilde{J}_3 &-\varkappa \widetilde{J}_2 \\ -\sigma \widetilde{x}_3 & 0 & \widetilde{x}_1 & -\varkappa \sigma \widetilde{J}_3 & 0 & \varkappa \widetilde{J}_1 \\ \widetilde{x}_2 & -\widetilde{x}_1 & 0 & \varkappa \widetilde{J}_2 & -\varkappa \widetilde{J}_1 & 0 \\ \end{pmatrix}.
\end{equation*}
\notag
$$
Below we drop ‘tildes’ in our notation for coordinates. Note that the law of motion of the dynamical system with Hamiltonian $H$ can also be written in the vector form as Euler’s pseudospherical equations:
$$
\begin{equation*}
\begin{gathered} \, \dot{\vec{J}}=(g \vec{J}) \times \frac{\partial H}{\partial \vec{J}}+(g \vec{x}) \times \frac{\partial H}{\partial \vec{x}}, \\ \dot{\vec{x}}=(g \vec{x}) \times \frac{\partial H}{\partial \vec{J}}. \end{gathered}
\end{equation*}
\notag
$$
The main idea is to find linear changes of variables in $\mathbb{C}^6$ (because the system can also be considered for complex coordinates $J_i$ and $x_i$) that transform $H$ and the Poisson bracket $P$ into real $\widetilde{H}$ and $\widetilde{P}$ again. Note that there need not exist a real change of variables $\mathbb{R}^6 \to \mathbb{R}^6$ for the resulting system that recovers the original form of the integral and bracket. If the original system has a first integral $K$, then after applying this change of variables to $K$ we still have involutivity: $\widetilde{P}(\widetilde{H}, \widetilde{K})=0$. In particular, if the skew gradients of $H$ and $K$ are linearly dependent at a point (so that this point is critical), then the skew gradients of their images are also linearly dependent at the image of this point. For many known mechanical systems the above transformation $\Phi$ leaves $K$ real. For instance, this is true for the Euler, Lagrange and Kovalevskaya tops (provided that we multiply the coefficient of $x_1$ in the Hamiltonian and the relevant coefficients in the Kovalevskaya integral by $i$), and the Goryachev-Chaplygin and Hess systems. Note that for the family of Kovalevskaya systems on the Lie algebras $\mathrm{so}(3, 1)- \mathrm{e}(3)- \mathrm{so}(4)$, which includes the classical top, this fact holds for any $\varkappa \in \mathbb{R}$. Under the transformation $\Phi$ introduced above the Hamiltonian and first integral of the Euler top are taken to the form
$$
\begin{equation}
H=\frac{J_1^2}{2A_1}+\frac{J_1^2}{2A_2}-\frac{J_3^2}{2A_3}=h
\end{equation}
\tag{1.1}
$$
and where the $A_i >0$ are the principal moments of inertia of the system. Here and below we assume that these moments of inertia are pairwise different. Casimir functions of the resulting bracket $P_{\mathrm{ps}}$ have the form Thus the problem of the investigation of the phase topology of Euler’s system has reduced to the description of common level sets of the four first integrals listed above and the description of their bifurcations
§ 2. Main resultsIn this paper we mostly describe the topology of the Liouville foliation of a pseudo-Euclidean analogue of Euler’s case for the following special values of the Casimir functions: we set $a\cdot b=0$, but exclude the case when $a=b=0$. Lemma 1. Let the pair $f_1$, $f_2$ take a value $(a, b)$ distinct from $(0, 0)$. Then $M^4_{a, b}$ is a smooth 4-manifold and a symplectic leaf of the Poisson bracket $P_{ps}$ for $\varkappa = 0$. Since for $(J_1, J_2, -J_3) \ne (0, 0, 0)$ the vectors $\mathrm{d}f_1$ and $\mathrm{d}f_2$ are linearly independent, while otherwise we have $f_1=f_2=0$, for any values $(f_1, f_2)=(a, b)$ distinct from $(0, 0)$ the common level $M^4_{a, b}$ is a smooth 4-manifold. In the next theorem we describe the topological types of leaves of the Liouville foliation in the pseudo-Euclidean Euler case for various values of the moments of inertia and the values of the Casimir functions as indicated above. Theorem 1. 1. The Liouville foliations on the common level sets $M_{a, b}^4$ for the pseudo-Euclidean analogue of the Euler top have the following regular 2-dimensional leaves in the case when $a \cdot b=0$ and $(a, b) \ne (0, 0)$: each leaf is either homeomorphic to a 2-torus $T^2$, a cylinder $\mathrm{Cyl}^2$, or a plane $R^2$ (leaves homeomorphic to a torus occur for $A_3<A_2<A_1$, while ones homeomorphic to the plane occur for ${A_2<A_3<A_1}$). 2. If $A_3<A_2<A_1$, then for the same values of $a$ and $b$ the Liouville foliations of the same system can have the following singularities, which are analogues of Fomenko 3-atoms. Bifurcations of regular leaves of the system occurring in the preimage of a small line segment intersecting the arc $\Sigma_{a, b}$ transversally have the type of a Cartesian product of the foliated 2-base by a fibre homeomorphic to an interval $I$. (In what follows the notation used for a singularity in this case involves ‘hats’.) On levels $M^4_{a, b}$ under consideration the system has singularities of one of the following three types.
Next we go over to the description of the rough topology on nonsingular isointegral surfaces $Q^3_k$ and nonsingular isoenergy surfaces $Q^3_h$ for the pseudo-Euclidean Euler top. Claim 1. Assume that a pseudo-Euclidean Euler system is specified by three pairwise distinct moments of inertia $A_1$, $A_2$ and $A_3$. Each nonsingular ($h \ne 0$) surface $Q^3_h$ on a symplectic leaf $M^4_{a, b}$ of this system, where $(a, b) \ne (0 ,0)$ and $a \cdot b=0$, belongs to at most one of six leafwise homeomorphism classes. A similar result also holds for nonsingular ($k \ne 0$) isointegral surfaces. This result is a consequence of the following observation. Recall that $f_1$ and $H$ (or $K$) are quadratic polynomials in $\vec{x}=(x_1, x_2, x_3)$ and $\vec{J}=(J_1, J_2, J_3)$, and $f_2$ is the scalar product of $\vec{x}$ and $\vec{J}$ in the metric $(1, 1, -1)$. Thus, if $a \cdot b=0$, then dilations/contractions of coordinates take $Q_{a, b, h}$ to the level set $Q_{\operatorname{sgn}a,\operatorname{sgn}b,\operatorname{sgn}h}$. The condition $(a, b) \ne (0 ,0)$ means that $M^4_{a, b}$ is a smooth manifold in $\mathbb{R}^6$ and a symplectic leaf of the Poisson bracket. As $a \cdot b \ne 0$, up to dilations/contractions of the triples $\vec{J}$ and $\vec{x}$, the following three cases are possible:
$$
\begin{equation*}
a=1, \quad b=0; \qquad a=-1, \quad b=0; \qquad a=0, \quad b=\pm 1.
\end{equation*}
\notag
$$
Each case corresponds to two classes of isoenergy surfaces $Q_h$ depending on the sign of $h \ne 0$ (the sign of $k$ for isointegral surfaces $Q_k$). Remark 1. For two systems with different triples of principal moments of inertia $A_1$, $A_2$ and $A_3$, these sets of six classes can be distinct. In this paper we consider the case when the principal moments of inertia $A_i$ are pairwise distinct and $A_3$ corresponds to the ‘negative’ axis of the pseudo-Euclidean space. The next observation shows that we only need to examine two cases: when $A_3$ is the maximum/minimum quantity among $A_1$, $A_2$, $A_3$ and when it is in the middle, for instance, $A_1>A_3 >A_2$. Lemma 2. The Hamiltonian $H$ with pairwise distinct moments of inertia ${0<A_3<A_1<A_2}$ defines the same Liouville foliation as the Hamiltonian $\widetilde{H}$ with moments of inertia $0< \widetilde A_1<\widetilde{A_2}<\widetilde A_3$. If $A_3$ is the greatest moment of inertia, then we subtract the Hamiltonian $H$ with a suitable coefficient $\alpha >0$ from the additional first integral $K=J_1^2+J_2^2-J_3^2$ and obtain the Hamiltonian
$$
\begin{equation*}
H=\biggl( 1 -\frac{\alpha}{2A_1}\biggr) J_1^2 +\biggl(1 -\frac{\alpha}{2A_2}\biggr) J_2^2 -\biggl(1 -\frac{\alpha}{2A_3}\biggr) J_3^2,
\end{equation*}
\notag
$$
where all coefficients are nonnegative and pairwise distinct, and the coefficient corresponding to the third, ‘negative’, axis is the greatest one. In a similar way a system with moments of inertia $A_1<A_3<A_2$ can be taken to a system with $A_2<A_3<A_1$. In Theorem 2 below we describe the structure of the base of the Liouville foliation on nonsingular levels $Q^3_h=Q^3_{a, b, h}$ and $Q^3_k=Q^3_{a, b, k}$ in the case when $A_3<A_2<A_1$. We denote the resulting $12$ types of nonsingular levels by symbols $h$ and $k$ (for isoenergy and isointegral surfaces, respectively) and numbers from $1$ to $6$. The indices $1$ and $2$ correspond to $a=1$ and $b=0$, $3$ and $4$ correspond to $a=-1$ and $b=0$, and $5$ and $6$ to $a=0$ and $b \ne 0$. In each pair of indices, the odd one corresponds to $h < 0$ or $k<0$, while the even one corresponds to $h>0$ or $k>0$. Theorem 2. Analogues of rough molecules (Fomenko invariants) for the Liouville foliations corresponding to the pseudo-Euclidean Euler top with moments of inertia $A_3<A_2< A_1$ on nonsingular levels of energy $Q_h=Q_{a, b, h}$ or of the additional integral $Q_k=Q_{a, b, k}$ are shown in Figures 3–5, depending on the signs of $f_1=a$ and $ f_2=b$. The notation $1k, \dots, 6k$ and $1h, \dots, 6h$ for the invariants is the same as the notation for nonsingular levels introduced above. Both the base of the foliation (up to a homeomorphism) and the lift can uniquely be recovered in a neighbourhood of each value of $h$ (for an isointegral surface $Q^3_k$) or $k$ (for an isoenergy surface $Q^3_h$) from the data presented. If an invariant is disconnected, then it is assumed that bifurcations occur simultaneously in all components, that is, for the same values of the function (namely, the energy or the additional integral) defining the foliation of $Q^3$. In the topological analysis of the pseudo-Euclidean Euler system we follow the scheme presented below:
In this way we construct the bifurcation diagrams for the moment map
$$
\begin{equation*}
\mathfrak{F}=(H, K)\colon M^4_{a, b} \to \mathbb{R}^2(h, k)
\end{equation*}
\notag
$$
for each of the cases listed above. Here is the final result. Theorem 3. The bifurcation diagrams $\Sigma_{a, b}$ and the image of the moment map $(H, K)$ of the system (as restricted to the symplectic leaves $M^4_{1, 0}$, $M^4_{-1, 0}$ and $M^4_{0, 1}$) are as shown in Figures 6–9. The diagram $\Sigma_{a, b}$ consists of the singular point $(0, 0)$ and the three rays $h=k/(2A_1)$, $h= k/(2A_2)$ for $ k >0$ or $h=k/(2A_3)$ for $k <0$, and $k=0$ for $h< 0$ (for $A_3<A_2<A_1$), or the two rays of the straight line $k=0$ that start at the origin (for $A_2<A_3<A_1$). For each stratum (a singular point, an open ray or a two-dimensional component bounded by two rays) two curves are indicated whose product corresponds to the leaf in question of the Liouville foliation. The first term corresponds to the topological type of the intersection of the levels $H=h$ and $K=k$ in the space $\mathbb{R}^3(J_1, J_2, J_3)$, and the second to the topological type of the intersection of the generalized hyperboloid $f_1=a$ and the plane $f_2=b$ (with coefficient vector $\vec{J}$ ) in the space $\mathbb{R}^3(x_1, x_2, x_3)$. Note that critical bifurcation atoms correspond to rays $h=k/(2A_i)$, while noncompact noncritical bifurcations correspond to rays lying on the $Oh$-axis. § 3. Auxiliary results and their proofsIn this section we present a number of auxiliary results, which yield Theorems 1–3. 3.1. The topology of common levels sets $H=h$, $K=k$ in $ {\mathbb{R}}^3(J_1, J_2, J_3)$We regard the Hamiltonian and the additional first integral of the system as quadrics in $\mathbb{R}^3(J_1,J_2, J_3)$. Fixing some level $K=J_1^2+J_2^2-J_3^2=k$ we consider the foliation defined by the function $H$ on this level. Going over from $H=h$ to the level $H-K/(2A_3)=h-k/(2A_3)$ of the function $H-K/(2A_3)$ simplifies the problem significantly because the intersection of the hyperboloid $K=k$ with the cylinder
$$
\begin{equation*}
\biggl(\frac{1}{2A_1}-\frac{1}{2A_3}\biggr) J_1^2 +\biggl(\frac{1}{2A_1}-\frac{1}{2A_3}\biggr) J_2^2=h-\frac{k}{2A_3}
\end{equation*}
\notag
$$
can be described as follows. Claim 2. The common level $H=h, K=k$ of $H$ and $K$ in $\mathbb{R}^3(J_1, J_2, J_3)$ (denoted by $\xi^1_{h, k}$) is symmetric under the reflections $J_1 \to-J_1$ and $J_2 \to-J_2$. Its homeomorphism classes for nonsingular values of $(h, k)$ and for $(h, k)=(0, 0)$, as well as the types of bifurcation atoms in the preimages of small line segments $h=h_{\mathrm{crit}} \pm \varepsilon$ for $k \ne 0$ are presented in Table 1. Table 1.The topology of regular common level sets $H=h$, $K=k$ in the three-dimensional velocity space for various $(h, k)$, and the types of bifurcations of these sets (2-atoms) on the rays $h=k/(2A_i)$ on the bifurcation diagram. Here $A$ and $C_2$ are 2-atoms as in [2], and $B''$ is a Morse saddle (a 2-atom $B$ with two ‘cuts’). The symbols $4I_{-}$ and $4I_{+}$ denote the connected components homeomorphic to a line segment and intersecting the plane $J_3=0$ or disjoint from it, respectively. The symbol $Z$ denotes the union of the four common generators of the two cones $K=0$ and $H=0$ in the case $A_2<A_3<A_1$
Proof. Consider the intersection of the cylinder $H-\alpha K=h-\alpha k$ with the generalized hyperboloid $K=k$.
For $A_3<A_2<A_1$ the intersection is compact because the cylinder has the generalized elliptic type (the quantities $1/A_3-1/A_1$ and $1/A_3-1/A_2$ have the same sign). Thus, regular leafs are empty of homeomorphic to a circle, and their bifurcations are 2-atoms (as follows from an easy nondegeneracy check). For $A_2<A_3<A_1$ the cylinder has the generalized hyperbolic type ($1/A_3-1/A_1$ and $1/A_3- 1/A_2$ have different signs). Its generator (a straight line parallel to the $OJ_3$-axis) intersects the hyperboloid $K=k$ in two points, has a single common points with it (for $J_3=0$), or is disjoint from it (when it intersects the $OJ_1J_2$-plane inside the ‘neck’ of the hyperboloid of one sheet). To prove this consider the mutual position of the cross-section of the cylinder by the $OJ_1J_2$-plane (which is a hyperbola or a pair of intersecting straight lines) and the projection of the generalized hyperboloid $K=k$ onto this plane. The image of this projection is the whole plane for $k \leqslant 0$ or the exterior of a disc $J_1^2+J_2^2 \geqslant k$, where $k >0$. If the cross-section of the cylinder attains the boundary of the image of the projection, then the variable $J_3$ assumes all real values on the connected component of the level set $\xi_{h, k}$. We denote such components by $I_{-}$. Otherwise (for instance, if both quadrics $H=h$ and $K=k$ are hyperboloids of two sheets) their intersection is disjoint from the plane $J_3=0$. We denote such connected components, homeomorphic to an interval, by $I_{+}$. The verification that the critical points other than $(0, 0, 0)$ are Morse is trivial. Thus, we have a disjoint union of two Morse saddles $B''$ on the corresponding level $H=h_{\mathrm{crit}}$. Note that critical points correspond to points of intersection of a hyperboloid of two sheets with the $OJ_3$-axis or a hyperboloid of one sheet with the $OJ_1$- and $OJ_2$-axes. The proof is complete. 3.2. Common level sets of Casimir functions as functions of the coordinates of the topThe assumption $a \cdot b=0$ simplifies significantly the problem of the description of the common level $f_1=a$, $f_2=b$ of the Casimir functions as quadrics in $\mathbb{R}^3(x_1, x_2, x_3)$ for fixed $\vec{J}=(J_1, J_2, J_3)$. Then the area integral $f_2=b$ corresponds to a plane in $\mathbb{R}^3(\vec{x})$, and the geometric integral defines the generalized hyperboloid $x_1^2+x_2^2-x_3^2=a$. Recall that the vector $\vec{n_e}$, which is orthogonal to the plane $f_2=b$ in the Euclidean sense, is parallel to the vector $(J_1, J_2, -J_3)$. When $b=0$, the plane passes through the origin in $\mathbb{R}^3(\vec{x})$, contains the radius vector $\vec{J}$ and is orthogonal to $(J_1, J_2, -J_3)$ in the Euclidean sense. In Table 2 we present information of the structure of the common level set $f_1=a$, $f_2=0$, $K(\vec{J})=k$ in its dependence on $a$ and $k$. Table 2.The common level set $f_1=a$, $f_2=0$ (for a fixed vector $\vec{J}=(J_1, J_2, J_3)$) it its dependence on the signs of $a$ and $K=k$ (the value of the first integral at $\vec{J}$)
When $f_1=a=0$ and $f_2=b \ne 0$, we have the intersection of a cone with a plane not containing the origin. Making contractions/dilations of the coordinates $x_1$, $x_2$ and $x_3$ we obtain a problem for $a=0$ and $b=1$ and the same values $h$ and $k$ of the functions $H$ and $K$ at the vector $\vec{J}=(J_1, J_2, J_3)$. If the plane $f_2=1$ is parallel to a generator of the cone (so that $\vec{J}$ is a light vector and $k=0$), then the intersection of the plane and cone is a parabola (the plane does not contain the origin). If $\vec{J}$ lies inside the light cone, then the intersection of the plane and the cone is an ellipse, and if $\vec{J}$ lies outside the cone, then we have the two branches of a hyperbola. Thus we have established Theorem 3 in the part concerning the arcs forming the bifurcations diagrams for various $a$ and $b$. 3.3. Two-dimensional leaves and neighbourhoods of singular leavesWe see form the above that each common level $f_1=a$, $f_2=b$, $H=h$, $K=k$ can be represented as a fibre bundle without singularities. Its base is the common level $\xi^1_{hk}=\{\vec{J}=(J_1, J_2, J_3)\mid H(\vec{J}\,)=h,\, K(\vec{J}\,)=k\}$ of $H$ and $K$ in the space $\mathbb{R}^3(J_1, J_2, J_3)$. The homeomorphism class of a fibre of this fibration is independent of the particular point on the level surface $K=k$ (in particular, on the quantity $H=h$) for fixed values $a$ and $b$, $a \cdot b=0$, of the Casimir functions and fixed $k$. If a common level of the four functions on $\mathbb{R}^6$ contains no critical points of the map $(f_1, f_2, H, K)$, then it is orientable. Hence, in particular, each regular leaf of the Liouville foliation is homeomorphic to a torus, a cylinder, or a plane. The proof that the preimage of each point in the $Ohk$-plane lying away from the bifurcation diagram is as indicated in Theorem 3, for systems with $A_2<A_3<A_1$ follows from the fact that for all $a$ and $b$ considered here each connected component of $\xi^1_{hk}$ is simply connected. For $A_3<A_2<A_1$ connected components of $\xi^1_{h, k}$ can be homeomorphic to a circle. Thus, the existence of a disconnected fibre (the intersection of $f_1=a$ and $f_2=b$ in $\mathbb{R}^3(\vec{x})$) over each point in the base $\xi^1_{h, k}$ does not mean in general that the common level of the four first integrals is disconnected. Nevertheless, we can deduce this on the basis of the nondegeneracy of critical points in the preimages of the rays $h=k/(2A_2)$, $k >0$, and $h=k/(2A_3)$, $k<0$. These critical points are of elliptic type, that is, in the preimage of a small transversal line segment (either $h=\mathrm{const}$ or $k=\mathrm{const}$) we have the Cartesian product of a 2-atom by a disjoint union of circles and intervals. Theorem 3 is proved. 
Citation in format AMSBIB
\Bibitem{AltKib23} |
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