V. V. Vedyushkina, S. E. Pustovoitov, “Classification of Liouville foliations of integrable topological billiards in magnetic fields”, Sb. Math., 214:2 (2023), 166

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Sbornik: Mathematics, 2023, Volume 214, Issue 2, Pages 166–196
DOI: https://doi.org/10.4213/sm9770e (Mi sm9770)

This article is cited in 3 scientific papers (total in 3 papers)

Classification of Liouville foliations of integrable topological billiards in magnetic fields

V. V. Vedyushkina, S. E. Pustovoitov

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

English version PDF (823 kB) HTML full-text Citations (3) Russian version article

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DOI: https://doi.org/10.4213/sm9770e

Abstract: The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations — such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere.
Bibliography: 13 titles.

Keywords: integrable systems, magnetic field, topological billiard, Liouville foliation, Fomenko-Zieschang invariant.

Funding agency	Grant number
Russian Science Foundation	20-71-00155
This research was supported by the Russian Science Foundation under grant no. 20-71-00155, https://rscf.ru/en/project/20-71-00155/.

Received: 04.04.2022

Bibliographic databases:

Document Type: Article

MSC: Primary 37С83; Secondary 37D40, 70E17

Language: English

Original paper language: Russian

§ 1. Introduction

One natural extension of integrable planar billiards is the construction of topological billiards (see [1]), which allows one to preserve the integrability of the system. A topological billiard table is an orientable manifold obtained by gluing elementary planar billiards isometrically along common boundaries. A point mass which occurs on a gluing edge of such a billiard continues its motion on another sheet of the billiard table upon reflection. As a rule, it is assumed that the projections of both sheets glued together onto the plane lie to one side of the projection of their common boundary segment, which is an arc of a curve in a confocal family of quadrics, namely, ellipses and hyperbolae. The main object of study in our work is the Liouville foliation, which is the partition of the relevant four-dimensional phase manifold into common level surfaces of two independent integrals of motion, functions preserving their values on trajectories. Recall that in the case of nonresonant smooth or real-analytic Liouville-integrable Hamiltonian systems every such surface is the closure of the union of trajectories corresponding to some level of the integrals and, under some additional conditions, almost all such surfaces are homeomorphic to a 2-torus.

In the recent paper [2] Mironov and Byaly proved that a planar billiard in a magnetic field is integrable if and only if it is bounded by concentric circles, with the possible exception of a finite number of values of magnetic induction. So, as elementary billiards (‘bricks’ used to glue the table, which is a 2-manifold) we take billiards in a disc bounded by a circle or in an annulus bounded by two concentric circles. It is obvious that by gluing such domains together we can obtain a billiard-table complex which is homeomorphic to a disc, an annuls (or cylinder), a sphere, or a torus. For each of these topological billiards we investigate the topology of the Liouville foliation arising on an isoenergy surface. Recall that the square of the radius of the Larmor circle plays the role of energy in such systems.

The main result in this paper is an algorithm calculating the Fomenko-Zieschang classifying invariant for each topological magnetic billiard of any of the four types above, and also constructing the corresponding bifurcation diagram. In addition, we calculate the topological types of isoenergy manifold for elementary magnetic billiards and discover systems of rigid body dynamics which are Liouville equivalent to them, such as the Euler, Lagrange, Kovalevskaya, Joukowsky, and other cases of rigid body dynamics.

Two phenomena occur here. First, in magnetic topological billiards the properties of trajectories can change depending on whether the gluing arcs are convex or not. It is known that in locally planar topological billiards without potential and magnetic field the presence of a nonconvex gluing edge (for example, when two equal annuli are glued along their inner circles) results in our inability to define consistently (by continuity) the extension of a trajectory for which the gluing circle is a caustic. On the other hand a convex gluing edge enables one to extend each trajectory. Nevertheless, we point out that the Liouville foliation is well defined in both cases (and its singular two-dimensional leaf is still homeomorphic — see [3] — to the singular leaf of the nondegenerate semilocal singularity of rank 1 defined by a Fomenko 3-atom — see [4]).

Now, when a magnetic field is added, the converse situation is also possible, when convex gluing curves do not necessarily mean a well-defined extension for some trajectories, whereas in a system with a nonconvex gluing edge we can now have this property. This is because in a magnetic billiard system trajectories are not line segments but rather arcs of circles. Whether or not a trajectory has the property of extension past a tangency to a gluing curve depends on the type of tangency, which can be from outside or from inside. Below we determine the topology of 3-atoms arising in a magnetic billiard system in its dependence on the shape of the table.

Second, in magnetic topological billiards we can realize an interesting class of singularities. Namely, on some isoenergy 3-surfaces we have found saddle singularities whose critical circles are oppositely oriented by the motion of the particle (that is, by the Hamiltonian energy flow of the system). In this case the Liouville foliation is defined, but the Fomenko-Zieschang invariant is not. This phenomenon occurs in some well-known problems in mathematical physics. This is possible when a Larmor circle is tangent, from inside and from outside, to two circles which are gluing arcs of the table complex, one of which is convex and the other nonconvex. An arbitrarily small change in the values of the integrals (the radii of the circle of centres and the Larmor circle) breaks this singularity.

Note that another class of integrable Hamiltonian systems with magnetic field was investigated by Kudryavtseva and Oshemkov [5], who considered the geodesic flow on a 2-surface of revolution in a magnetic field. They considered systems on a surface homeomorphic to a sphere (in this paper we consider locally planar tables homeomorphic to a torus, a sphere, a disc or an annulus) with a nonflat metric. It turns out that nondegenerate split singularities also arise in that case.

§ 2. Requisite definitions

Recall some requisite definitions. A smooth manifold $M^4$ is said to be symplectic if a symplectic structure, which is a nondegenerate closed 2-form $\omega$ , is defined on it. By the skew gradient of a smooth function $H$ we mean the vector field $\operatorname{sgrad} H=\Omega^{-1} \operatorname{grad} H$ , where $\Omega$ is the matrix of $\omega$ in local coordinates. A dynamical system $v=\operatorname{sgrad} H$ is called a Hamiltonian dynamical system with Hamiltonian $H$ . If a Hamiltonian system $v$ on $M^{4}$ has two functionally independent functions $f_1$ and $f_2$ which are first integrals of it (that is, preserve their values along trajectories of the system, which is equivalent to $\{H, f_i\}=0$ ) and are in involution, that is, $\{f_1, f_2\}=0$ , and if their Hamiltonian vector fields $\operatorname{sgrad} f_i$ are complete, then such a system is said to be completely Liouville integrable (see [4]). The Liouville foliation is the partition of $M^4$ into connected components of common level surfaces of the integrals $f_1$ and $f_2$ . It is easy to see that any such surface is invariant under the flow of $v$ . By Liouville’s theorem a regular compact connected component of a common level surface of $f_1$ and $f_2$ is diffeomorphic to a torus $T^2$ . It is called a Liouville torus. In a neighbourhood of a Liouville torus in $M^4$ the Liouville foliation is diffeomorphic to the Cartesian product of a torus and a disc $D^2$ . In what follows we assume that $f_1$ coincides with $H$ , the energy or Hamiltonian of the system. We also assume that Bott’s condition holds: critical points of $f_2$ on $Q^3=\{x\in M^4\colon H(x)=\mathrm{const}\}$ must be nondegenerate and form critical submanifolds. In addition, we assume that all these manifolds are homeomorphic to circles. Consider a nonsingular (that is, $dH\neq0$ ) isoenergy manifold $Q^3$ . Several equivalence relations between systems restricted to a nonsingular energy level are used in the theory of integrable Hamiltonian systems. One of the most important questions is the equivalence of systems from the standpoint of leafwise homeomorphism of the Liouville foliations corresponding to two systems (note that other equivalence relations, when the $Q^3$ under consideration are homeomorphic or the orbital equivalence of integrable systems are also of considerable interest; see [4]). More precisely, we mean the following equivalence relation: two systems $v_1$ and $v_2$ on $Q_1^3$ and $Q_2^3$ , respectively, are said to be Liouville equivalent if there exists a leafwise diffeomorphism from $Q_1^3$ to $Q_2^3$ that preserves the orientations of these manifolds and the orientations of the critical circles. In other words, since almost all tori are nonresonant (that is, the winding trajectory are dense on them), the closure of a solution is a Liouville torus. So we can say that Liouville equivalence is the equivalence of the closures of almost all trajectories of the dynamical systems.

Next we describe briefly the construction of the classifying invariant of Liouville equivalence of systems, the Fomenko-Zieschang invariant (or marked molecule). This question was described in detail in [4], vol. 1, Ch. 4. Let us foliate a nonsingular isoenergy manifold $Q^3$ by level surfaces of the additional integral $f_2$ . Recall that by Liouville’s theorem a typical common level set of the energy $H$ and the integral $f_2$ (on level sets where $\operatorname{sgrad} H$ and $\operatorname{sgrad} f_2$ are linearly independent) is homeomorphic to a torus or a disjoint union of tori. Usually, a completely integrable system also has a finite number of singular level sets on a compact surface $Q^3$ . Under the assumption that $f_2$ is Bott on $Q^3$ , invariant neighbourhoods of singular level sets were classified by Fomenko (see [4], vol. 1, Ch. 3) up to leafwise homeomorphisms. Classes of leafwise homeomorphic small invariant neighbourhoods of such singular levels are called 3-atoms. To an isoenergy manifold and the Liouville foliation on it we can assign its Fomenko invariant (rough molecule), which is a Kronrod-Reeb graph such that the interior points of its edges correspond to regular leaves (that is, Liouville tori) in $Q_3$ , and vertices correspond to atoms, that is, Bott bifurcations, of Liouville tori. The Fomenko-Zieschang invariant (marked molecule) is a rough molecule some subgraphs of which are assigned numerical marks describing the correspondence (that is, gluing homeomorphism) between the boundary tori of the two atoms connected by the edge in question.

Theorem 1 (Fomenko and Zieschang; see [4]). Two systems $v_1$ and $v_2$ on $Q_1^3$ and $Q_2^3$ are Liouville equivalent if and only if their marked molecules coincide.

For more details on the classifications of atoms and on integrable Hamiltonian systems the reader can consult [4] and also the original papers [6] and [7]. For a circumstantial account of applications of the classification theory developed to various problems in mathematical physics, see [8]–[11].

§ 3. Statement of the problem. Trajectories

Consider a billiard system inside a circle. When a point mass occurs at the boundary of the billiard table, it bounces off it absolutely elastically, so that the angle of incidence is equal to the angle of reflection and the modulus of the velocity remains the same. Assume that a constant magnetic field with constant induction $b$ acts on the unit point mass with unit charge. Then it is known that, between reflections from the side of the table the point mass moves along an arc of a circle of fixed radius $L$ anticlockwise (this circle is called a Larmor circle). The modulus of its velocity remains the same and is equal to $bL$ , and the Hamiltonian of this system, its kinetic energy, depends quadratically on $L$ . In what follows, in our investigations of the Liouville foliation, we use the first integral $L$ in place of this Hamiltonian.

We can also show that during the motion (between and after reflections) the distance from the centre of the boundary circle (the centre of the table) to the centre of the corresponding Larmor circle remains the same. Let $R$ denote this distance. Analytically, this function has the following form:

$\begin{equation} R=\frac{1}{b}\sqrt{\dot{x}^2+\dot{y}^2+b^2(x^2+y^2)-2b(x\dot{y}-y\dot{x})}. \end{equation} \tag{3.1}$

Hence the trajectory of the motion of a point mass inside a circular billiard domain is piecewise smooth and consists of arcs of Larmor circles with the same radius

$L$ , whose centres lie at the same distance

$R$ from the centre of the table (Figure 1, a).

Figure 1.Examples of trajectories of motion for a circular and an annulus-like magnetic billiards.

Note that a billiard in a domain bounded by two concentric circles has the same first integrals $L$ and $R$ , and the trajectory of motion also consists of arcs of circles of the same radius, whose centres lie at the same distance from the centre of boundary circles (Figure 1, b).

Lemma 1. The functions $L$ and $R$ are first integrals of the magnetic billiard in a circular (annulus-like) domain. These functions are functionally independent almost everywhere and are in involution with respect to the standard symplectic structures on $M^4$ . Furthermore, the integral $R$ is a Bott function (that is, the critical submanifolds of $R$ are nondegenerate). Thus, the magnetic billiard in a circular (annulus-like) domain is Liouville integrable.

§ 4. Regions of possible motion. Bifurcation diagrams

Definition 1. For particular levels of the integrals $L$ and $R$ the domain of possible motion is the image of the corresponding leaf of the Liouville foliation under the projection $p\colon M^4\to \mathbb{R}^2$ onto the billiard table. In other words, it is the closed subdomain of the table in which a material point can occur for the prescribed values of the integrals $L$ and $R$ .

Now we find all domains of possible motion corresponding to various values of the parameters $L$ and $R$ for a circular and an annulus-like billiard table.

We start with a circular billiard. Let $R_0$ be the radius of the boundary circle. Fix some $L$ , $0<L<\frac{1}{2}R_0$ . For $R=0$ the domain of possible motion is one-dimensional: it consists of a Larmor circle of radius $L$ (Figure 2, a).

For $0<R< L$ trajectories of motion are closed Larmor circles lying fully inside the billiard domain and encircling the centre of the domain. Hence the domain of possible motion is an annulus bounded by the circles with radii $L-R$ and $R+L$ , and the particle moves anticlockwise (Figure 2, b).

For $L=R$ trajectories are still Larmor circle which now pass through the centre of the table. The domain of possible motion is the disc bounded by the circle of radius $R+L$ (Figure 2, c).

For $L<R\leqslant R_0-L$ trajectories are Larmor circles too. The domain of possible motion is the annulus bounded by the circles of radii $R-L$ and $R+L$ (Figure 2, d). Note that all trajectories are closed up to $R=R_0-L$ inclusive (in accordance with the above definitions, this means that all tori are resonant).

For $R_0-L<R<L+R_0$ the trajectory is piecewise smooth and consists of arcs of Larmor circles. The domain of possible motion is the annulus bounded by the circles of radii $R-L$ and $R_0$ ; the particle moves anticlockwise in this domain (Figure 2, e).

For $R=L+R_0$ the domain of possible motion degenerates into the circle of radius $R_0$ , and then for $R>L+R_0$ the Larmor circles do not intersect the billiard domain and the domain of possible motion is an empty set (Figure 2, f).

Figure 2.Regions of possible motion for $0<L<\frac{1}{2}R_0$ .

Now fix $L$ , $\frac{1}{2}R_0\leqslant L<R_0$ . For $0<R\leqslant R_0-L$ trajectories of motion are closed Larmor circles which lie fully inside the billiard domain. Hence the domain of possible motion is the annulus bounded by the circles of radii $L-R$ and $R+L$ ; the particle moves anticlockwise in this domain (Figure 3, b).

For $R_0-L<R<L$ the trajectory is piecewise smooth and consists of arcs of Larmor circles. The domain of possible motion is the annulus bounded by the circles of radii $L-R$ and $R_0$ ; the particle moves anticlockwise in it (Figure 3, c).

For $L=R$ each arc of Larmor circles passes through the centre of the billiard table. The domain of possible motion is the whole table (Figure 3, d).

For $L<R<L+R_0$ the domain of possible motion is the annulus bounded by the circles of radii $R-L$ and $R_0$ ; the particle moves clockwise in it (Figure 3, e).

For $R=L+R_0$ the domain of possible motion degenerates into the circle of radius $R_0$ , and then, for $R>L+R_0$ , Larmor circles do not intersect the billiard domain and the domain of possible motion is an empty set (Figure 3, f).

Figure 3.Regions of possible motion for $\frac{1}{2}R_0\leqslant A<R_0$ .

Finally, fix $L\geqslant R_0$ . For $R<L-R_0$ Larmor circles do not intersect the billiard domain, and the domain of possible motion is empty.

For $R=L-R_0$ the domain of possible motion is one-dimensional; it is the boundary circle of the table (Figure 4, a).

For $L-R_0<R<L$ the trajectory is piecewise smooth and consists of arcs of Larmor circles. The domain of possible motion is the annulus bounded by the circles of radii $L-R$ and $R_0$ ; the particle moves anticlockwise in it (Figure 4, b).

Figure 4.Regions of possible motion for $L\geqslant R_0$ .

As the first integral $R$ grows further, trajectories in the domain of possible motion reproduce the case when $\frac{1}{2}R_0\leqslant L <R_0$ .

Lemma 2. The preimage under the projection $p$ of any two-dimensional domain of possible motion of a circular magnetic billiard system (that is, the corresponding leaf of the Liouville foliation) is homeomorphic to a $2$ -torus in $M^4$ .

Proof. First we consider a domain of possible motion not containing the centre of the billiard table and with boundary not containing the boundary of the table (Figure 5, a). Each point in the interior of this domain belongs to two Larmor circles. Hence there are two velocity vectors corresponding to each point. In a similar way, a unique velocity vector corresponds to each boundary point. We partition the domain of possible motion into radial segments and assign the corresponding velocity vectors to each point. The preimage in

$M^4$ of each segment is obviously homeomorphic to a circle. Hence the preimage of the whole domain is homeomorphic to a 2-torus.

Now consider a domain of possible motion containing the centre of the table in its interior, but with boundary not containing the boundary of the table either (Figure 5, b). As in the preceding case, a unique velocity vector corresponds to each boundary point, and two velocity vectors correspond to each interior point, except for a whole circle of vectors corresponding to the centre of the billiard table. We partition the domain of possible motion into diameters and assign the corresponding velocity vectors to points on them, while to the midpoint of each diameter we assign two vectors parallel to this diameter. It is easy to see that the preimage in $M^4$ of each diameter is homeomorphic to a circle. Hence the preimage of the whole domain of possible motion is homeomorphic to a cylinder with boundary circles glued in a certain way so that their orientations agree. Hence we obtain a 2-torus again.

Figure 5.

The remaining cases of domains of possible motions with boundaries containing the boundary of the billiard domain can be treated similarly. The only difference is that two velocity vectors, rather than one, correspond to points on the boundary of the table. However, these two vectors are identified in accordance with the law of refection.

The proof is complete.

Thus, for $0<L<R_0$ and $0<R<L+R_0$ , as well as for $L\geqslant R_0$ and $L-R_0<R<L+R_0$ , the inverse image of the domain of possible motion is homeomorphic to a 2-torus.

Definition 2. The moment map of an integrable system is the map $F\colon M^{4} \to \mathbb R^2$ such that $F(x)=(L(x), R(x))$ . The bifurcation diagram $\Sigma(F)$ is the image in $\mathbb R^2$ of the critical points of the moment map. Here a point in $M^4$ is said to be critical if the rank of the differential of the moment map drops at this point. Otherwise the point is regular.

In our case the image of the moment map is shown in Figure 6. The boundary of this domain is the bifurcation diagram.

Figure 6.The bifurcation diagram of a circular magnetic billiard.

Now consider a billiard in an annulus. Let $R_0$ be the radius of the outer circle and $r_0$ be the radius of the inner one. We reason in a similar way. Fix $L<r_0$ . For $R<r_0-L$ Larmor circles are disjoint from the billiard domain, and the domain of possible motion is empty. For $R=r_0-L$ the domain of possible motion is one-dimensional: it is the circle of radius $r_0$ (Figure 7, a).

For $r_0-L<R<L+r_0$ the trajectory is piecewise smooth and consists of arcs of Larmor circles. The domain of possible motion is the annulus bounded by the circles of radii $r_0$ and $R+L$ ; the particle moves anticlockwise in this domain (Figure 7, b).

For $L+r_0<R<R_0-L$ trajectories are Larmor circles which lie fully in the billiard domain. The domain of possible motion is the annulus bounded by the circles of radii $R-L$ and $R+L$ (Figure 7, c).

For $R_0-L<R<L+R_0$ trajectories become piecewise smooth again. The domain of possible motion is the annulus bounded by the circles of radii $R-L$ and $R_0$ , and the particle moves clockwise in this domain (Figure 7, d). For $R=L+R_0$ the domain of possible motion degenerates into a one-dimensional circle, and for larger values of $R$ it becomes an empty set.

Figure 7.Regions of possible motion for $L<r_0$ .

Now consider the case when $r_0<L<\frac{1}{2}(r_0+R_0)$ . For $R=0$ the domain of possible motion is a Larmor circle of radius $L$ (Figure 8, a).

For $0<R\leqslant L-r_0$ trajectories are Larmor circles lying fully in the billiard domain. The domain of possible motion is the annulus bounded by the circles of radii $L-R$ and $L+R$ , and the particle moves anticlockwise in this domain (Figure 8, b).

For $L-r_0<R\leqslant R_0-L$ trajectories become piecewise smooth. The domain of possible motion is the annulus bounded by the circles of radii $r_0$ and $R+L$ , and the particle moves anticlockwise in it (Figure 8, c).

For $R_0-L<R<L+r_0$ the point mass hits both inner and outer sides of the billiard domain. The domain of possible motion is the whole billiard domain and the motion in it changes from anticlockwise to clockwise one (Figure 8, d, e, f).

For $L+r_0\leqslant R<L+R_0$ the point mass hits only the outer side. The domain of possible motion is bounded by the circles of radii $R-L$ and $R_0$ , and the mass moves clockwise in it (Figure 8, g).

For $R=L_0+R$ the domain of possible motion degenerates into the circle of radius $R_0$ , and for greater values of $R$ it becomes an empty set (Figure 8, h).

Figure 8.Regions of possible motion for $r_0<L<\frac{1}{2}(r_0+R_0)$ .

Now consider the case when $\frac{1}{2}(R_0+r_0)<L<R_0$ . For $0<R\leqslant R_0-L$ trajectories are Larmor circles lying fully in the billiard domain. The domain of possible motion is the annulus bounded by the circles of radii $L-R$ and $R+L$ , and the particle moves anticlockwise in it (Figure 9, b).

For $R_0-L<R\leqslant L-r_0$ trajectories become piecewise smooth. The domain of possible motion is the annulus bounded by the circles of radii $R_0$ and $L-R$ , and the particle moves anticlockwise in this domain too (Figure 9, c).

For $L-r_0<R<L+r_0$ the point mass hits both inner and outer sides of the billiard table. The whole table is the domain of possible motion, and the motion in it changes from anticlockwise to clockwise (Figure 9, d, e, f).

For $L+r_0\leqslant R<L+R_0$ the point mass hits only the outer side. The domain of possible motion is bounded by the circles of radii $R-L$ and $R_0$ , and the particle moves clockwise (Figure 9, g) in it.

Figure 9.Regions of possible motion for $\frac{1}{2}(R_0+r_0)<L<R_0$ .

Finally, fix some $L>R_0$ . Then for $R<L-R_0$ Larmor circles do not intersect the billiard domain, and the domain of possible motion is empty. For $R=L-R_0$ the domain of possible motion is the circle of radius $R_0$ .

For $L-R_0<R<L-r_0$ trajectories are piecewise smooth and consist of arcs of Larmor circles. The domain of possible motion is the annulus bounded by circles of radii $R_0$ and $L-R$ , and the particle moves anticlockwise in this domain (Figure 10, b). As the value of $R$ grows further, the domains of possible motion and the behaviour of trajectories are just as in the case $\frac{1}{2}(R_0+r_0)<L<R_0$ , which we considered above.

Figure 10.Regions of possible motion for $L>R_0$ .

A lemma similar to Lemma 2 holds for annulus-like billiards.

Lemma 3. The preimage under the projection $p$ of any two-dimensional domain of possible motion of an annulus-like magnetic billiard (that is, the corresponding leaf of the Liouville foliation) is diffeomorphic to a $2$ -torus in $M^4$ .

Similarly to circular billiards, we can construct the image of the moment map and the bifurcation diagram for an annulus-like billiard (Figure 11).

Figure 11.The bifurcation diagram of an annulus-like magnetic billiard.

§ 5. Fomenko-Zieschang invariants. Isoenergy manifolds

Now we investigate the topology of the Liouville foliation of an isoenergy manifold $Q^3=\{x\in M^4\colon L(x)=\mathrm{const}\}$ . By Lemmas 2 and 3 regular leaves of the Liouville foliation are homeomorphic to 2-tori. Note also that for each value of $L$ only two critical leaves exist for circular and annulus-like billiards, both of which are homeomorphic to a circle. Their neighbourhoods are homeomorphic to a 3-atom $A$ . Hence the rough molecule corresponding to any regular value of the first integral $L$ has the form $A$ – $A$ .

Proposition 1. For a circular magnetic billiard the isoenergy manifold $Q^3$ is homeomorphic to a 3-sphere $S^3$ for any value of the first integral $L$ . The corresponding Fomenko-Zieschang invariant has the form $A$ – $A$ with marks $r=0$ and $\varepsilon=1$ .

Proof. Consider the case when

$0<R<\frac{1}{2}R_0$ . We partition

$Q^3$ into the two solid tori corresponding to

$0<R<L$ and

$L<R<L+R_0$ . They are glued along the boundary leaf

$L=R$ . In Figure 12 we show the projection of this boundary leaf onto the billiard table and the projections of regular leaves in neighbourhoods of the atoms

$A$ , with admissible basis cycles

$\lambda$ and

$\mu$ on these leaves. Note that the cycles

$\gamma$ shown in Figure 12, a and c, are glued to the cycle

$\gamma$ on the boundary leaf (Figure 12, b). In addition,

$\gamma$ on leaf ‘a’ is homologous to

$\pm\lambda_{-}+\mu_{-}$ , while on leaf ‘c’ it is homologous to

$\lambda_{+}$ . Furthermore, the cycle

$\mu_{+}$ is glued to

$-\mu_{-}$ because these cycles have opposite orientations (Figure 2). Hence the gluing matrix for basis cycles has the form

$\begin{pmatrix}1 & 1\\0 & -1 \end{pmatrix}$ . Such a gluing matrix corresponds to the marks

$r=0$ and

$\varepsilon=1$ , and the manifold

$Q^3$ is homeomorphic to the sphere

$S^3$ .

Figure 12.Basis cycles in a neighbourhood of a level $L=R$ .

Similar reasonings apply to $\frac{1}{2}R_0<A<R_0$ and $L>R_0$ , provided that we choose the cycles $\gamma$ in a suitable way.

The proof is complete.

Proposition 2. For an annulus-like magnetic billiard, for any value of $L$ the isoenergy manifold $Q^3$ is homeomorphic to the product $S^1\times S^2$ . The corresponding Fomenko-Zieschang invariant has the form $A$ – $A$ with marks $r=\infty$ and $\varepsilon=1$ .

Proof. It is easy to see from Figures 7–10 that for various values of the first integral

$L$ the basis cycles

$\lambda$ and

$\mu$ corresponding to neighbourhoods of the two atoms

$A$ are glued in accordance with the formulae

$\lambda_{+}=\pm\lambda_{-}$ and

$\mu_{+}=-\mu_{-}$ (this is because, in contrast to a circular billiard, for a billiard system in an annulus the domains of possible motion cannot transform past the centre of the table). Hence the gluing matrix is

$\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}$ . The marks are

$r=\infty$ and

$\varepsilon=1$ , and the isoenergy manifold

$Q^3$ is homeomorphic to

$S^1\times S^2$ .

The proof is complete.

Remark 1. Thus we have completed the topological analysis of integrable magnetic billiards on planar (elementary) billiard tables. We have calculated their marked molecules for each value of the energy and found the homeomorphism classes of the corresponding isoenergy manifolds $Q^3$ .

§ 6. Liouville equivalent systems

Now we show which known integrable systems with two degrees of freedom are Liouville equivalent to integrable planar magnetic billiards. To do this we use the Fomenko-Zieschang theorem stated above.

Proposition 3. A list of Liouville equivalences discovered between planar magnetic billiards and known integrable cases in rigid body dynamics is presented in Table 1. Figures in parentheses indicate the levels of energy on which the Liouville equivalence with billiards is known.

Table 1.Systems in rigid body dynamics which are equivalent to magnetic planar circular billiards (figures in parentheses denote the corresponding energy zones)

Cases of magnetic billiards	Molecule	Cases of rigid body dynamics
Circular billiard		Lagrange (1), Kovalevskaya (1), Joukowsky (7), Euler (1), Goryachev-Chaplygin-Sretenskii (1)
Ring billiard		Lagrange (2), Joukowsky (1)

Recall that a list of marked molecules calculated for many integrable systems in mechanics and mathematical physics can be found in [4], vol. 2.

§ 7. Isointegral manifold $Q^3_R$

Now we consider the isointegral manifold $Q^3_R=\{x\in M^4\colon R(x)=\mathrm{const}\}$ and carry out a similar analysis for it. We start with a circular billiard. In Figures 13–15 we show the transformations of domains of possible motion for fixed values of the first integral $R$ and an increasing value of $L$ . Note that, as with the isoenergy manifold $Q^3$ , the corresponding rough molecule has the form $A$ – $A$ . As before, $r=0$ , but for $0<R<R_0$ the orientation of the cycle $\mu_{-}$ in a neighbourhood of a minimal atom $A$ is not defined, so the mark $\varepsilon$ is not defined either. Nevertheless, $Q^3_R$ is also homeomorphic to the sphere $S^3$ .

Figure 13.The transformation of the domain of possible motion for $0<R<\frac{1}{2}R_0$ .

Figure 14.The transformation of the domain of possible motion for ${\frac{1}{2}R_0<R<R_0}$ .

Figure 15.The transformation of the domain of possible motion for $R_0<R$ .

Next consider an annulus-like billiard. We show the transformations of domains of possible motions for various values of the first integral $R$ in Figures 16–19. As in the previous cases, the rough molecule is $A$ – $A$ . For $r_0<R<R_0$ the gluing matrix is $\begin{pmatrix}\pm1 &0\\0 &\mp1\end{pmatrix}$ , where the sign cannot be determined because the orientation of the cycle $\mu_{-}$ is not defined. Hence $r=\infty$ , but the mark $\varepsilon$ is not defined. The isointegral manifold is homeomorphic to the product $S^1\times S^2$ .

Figure 16.The transformation of the domain of possible motion for $0<R<r_0$ .

Figure 17. The transformation of the domain of possible motion for $r_0< R< \frac{1}{2}(R_0+r_0)$ .

Figure 18. The transformation of the domain of possible motion for $\frac{1}{2}(R_0+r_0)< R<R_0$ .

Figure 19.The transformation of the domain of possible motion for $R_0<R$ .

For $R>R_0$ the cycles $\mu$ have opposite orientations. Hence the gluing matrix is $\begin{pmatrix} 1 &0\\0 &-1\end{pmatrix}$ . The marks are $r=\infty$ and $\varepsilon=1$ .

For $R<r_0$ the cycles $\mu$ have the same orientation. Hence the gluing matrix is $\begin{pmatrix} -1 &0\\ 0 & 1\end{pmatrix}$ . The marks are $r=\infty$ and $\varepsilon=-1$ .

§ 8. Topological billiards

We saw in the previous sections that the class of integrable planar magnetic billiards is quite small (it includes only circular and annulus-like billiards). Thus the number of types of Liouville foliations of isoenergy manifolds of such billiards is also small. The implies that such ‘simple’ billiards can realize just few of the known dynamical systems in the sense of Liouville equivalence (see Table 1). To enrich the class of magnetic billiards (or in other words, to realize much more integrable systems described, for instance, in [8]–[11]), we must generalize the concept of magnetic billiard. One such generalization of integrable planar billiards is the concept of topological billiard introduced by Vedyushkina [1]. It was originally introduced for classical billiard domains bounded by arcs of confocal hyperbolae and ellipses. If we take two such domains both of whose boundaries contain the same segment of a hyperbola or an ellipse, then we can glue these billiard domains along this boundary segment. Moreover, for simplicity we can always assume that the billiard domains glued together lie to the same side of this segment. A point mass moving in one billiard domain, after bouncing off the gluing segment, continues its motion in the other domain. As a result, we define a topological billiard as an orientable manifold glued in the above sense from integrable planar billiards belonging to the same class of integrability. For more information about gluings and conditions on them, see Vedyushkina [12].

Such a construction, first, preserves integrability and, second, extends significantly the class of integrable planar billiards bounded by arcs of confocal quadrics (recall that there exist only finitely many inequivalent planar billiards: see [1]).

We can also use this construction in our case. As integrable elementary billiards from which we glue a topological billiard, we take billiards of just two types considered above, a billiard domain bounded by a circle and a domain bounded by two circles. The piecewise planar orientable billiard-table complex obtained by such gluings is obviously homeomorphic to one of the following 2-manifolds: a disc, a cylinder, a sphere or a torus. Moreover, in such a table we can always distinguish a cylinder glued from annuli, whose boundary components are either not glued to anything, or a glued between themselves (so that we obtain a torus), or are glued up by one or two discs. In what follows we will work with these cylinders from the outset.

For more details, consider several annulus-like billiard tables numbered by positive integers from 1 to $n$ . Also assume that any two adjacent tables have outer or inner boundaries of the same radius, which we identify. In other words we glue the billiard tables under consideration along common boundary components, assuming additionally that no three tables are glued along the same boundary component. We obtain a complex homeomorphic to the cylinder $[0,1]\times S^1$ . We define billiard motion on each table as before, except that, when the point mass occurs at the boundary of a table, it goes over to the adjacent table with the same boundary and continues its motion along it. The billiard $bC$ thus constructed is a topological billiard. We present an example in Figure 20.

Figure 20.A topological billiard $bC$ on a table homeomorphic to a cylinder. The trajectory shown starts in the first annulus and, after bouncing off several times, occurs in annulus no. 4.

The billiard $bC$ has the same first integrals $L$ and $R$ , that is, it is Liouville integrable. Note that for fixed values of these integrals, on each elementary ‘sheet’ the domain of possible motion is also the intersection of the billiard domain and the domain $W$ bounded by the circles of radii $R+L$ and $|R-L|$ (we call it the initial domain of possible motion). Note that this domain of possible motion has several connected components, which are homeomorphic to cylinders. The preimage in the phase space $M^4$ of each connected component of a regular domain of possible motion is again homeomorphic to a 2-torus. A proof can be as follows. We cut the Liouville tori corresponding to each annulus along the points lying over the gluing edges. These cuts transform each torus into a ring. By gluing these ‘Liouville rings’ into a cylinder we arrange them into a single torus again. However, bifurcations of such tori and the 3-atoms describing them have now a more complex form.

Definition 3. Let $\tau$ be a finite string of positive integers. Let $n=\sum \tau_i$ . We denote by $B_\tau$ the saddle 3-atom of the type of a Cartesian product of a circle and a base (a 2-atom) of the following form: we cut a 2-atom $B_n$ transversally into $|\tau|$ connected components so that for each $i$ the $i$ th part contains precisely $\tau_i$ critical points, and we glue back these parts with twisting along the cuts (Figure 21). From the resulting 2-atom, by multiplying it by a circle, we obtain a 3-atom. Such a 3-atom describes a bifurcation of $k=\sum_{2\mid i}\tau_i+1$ Liouville tori into $l=\sum_{2\nmid i}\tau_i+1$ Liouville tori.

Figure 21.The 2-atom $B_\tau$ for $\tau=(2,2,1,1)$ .

Note that for each $\tau$ the 2-atom $B_\tau$ is planar (one possible embedding in the plane is shown in Figure 22); it also has the form shown in Figure 23, d. Also note that the class of 3-atoms $B_\tau$ is precisely the class of atoms $V_n^{\eta_1,\dots,\eta_n}$ (so-called atoms with pluses and minuses) appearing in [5] in the description of flows on surfaces of revolution in a magnetic field. We use other notation because the atom $V_n^{0,0,\dots,0}$ coincides with $B_n$ and because it will be more convenient to work with sequences $\tau$ .

Figure 22.The embedding of the 2-atom $B_\tau$ in the plane.

Figure 23.The construction of a rough molecule.

Now we discuss an algorithm that, given a topological billiard $bC$ , produces the rough molecule corresponding to some level of the integral $L$ .

Step 1. We number the gluing edges bearing in mind the elementary tables (sheets) containing them, so that the boundary components of a sheet must be numbered by consecutive positive integers.

Step 2. On the plane $(0xy)$ we draw a polygonal line ( $K_0$ in Figure 23, a) as follows: we join sequentially the points with coordinates $(i, R_i)$ , where $R_i$ is the radius of the $i$ th boundary component. Then we mirror reflect the resulting graph in the $0x$ -axis (the line $M$ in Figure 23, a). In other words, the curves $K_0$ and $M$ make up the profile of the billiard (its cross-section by a plane containing the symmetry axis of this topological billiard). Finally, we mirror reflect in the line $y=L$ the part of $K_0$ lying over this line (the polygonal line $K$ shown in Figure 23, a).

Step 3. We partition the domain between the lines $K$ and $M$ into horizonal line segments and contract each segment to a point. In the resulting graph we assign the index $A$ to free vertices. Note that any other vertex of the graph corresponds to the several local minima on the curve $K$ and local maxima on $M$ that lie on the corresponding horizontal line segment. Let the sequence of these extrema be described from left to right by a sequence $\tau$ (so that $\tau_1$ minima follow in a row, then $\tau_2$ maxima follow in a row, and so on). We present an example of such a sequence of extrema for $\tau=(2,2,1,1)$ in Figure 23, c. We assign the index $B_\tau$ to such a vertex.

Lemma 4. The graph with indexed vertices constructed above (Figure 23, b) is the rough molecule of the topological billiard $bC$ for the prescribed energy level $L$ .

Proof. Note that extrema on the polygonal curve

$K$ belong to one of four types. The first type is maxima occurring on the line

$y=L$ . A motion on the corresponding energy level

$R=0$ is a critical motion along Larmor circles on different elementary tables (sheets). As the value of the integral grows slightly, these circles, regarded as domains of possible motion, thicken into annuli (see Figure 24, a). Hence an atom

$A$ corresponds to each such point. The maxima on

$K$ that originally, on the curve

$K_0$ , lay below the line

$y=L$ can also be assigned to this type. They correspond to convex gluings of billiard tables along circles of radius less than that of Larmor circles. Hence a motion in a neighbourhood of such gluings is possible once the inner boundary circle of radius

$|L-R|$ of the domain

$W$ (the original domain of possible motion) attains the circle corresponding to the gluing. The further growth of

$R$ results in a motion involving two glued billiard domains (sheets) (see Figure 24, b).

Figure 24.Transformations of domains of possible motion: (b) two billiard tables (shown locally) glued along the convex (outer) part of the boundary; (c) gluing two billiard tables along the nonconvex (inner) part of the boundary.

The second type is maximum points on $K$ that are local minima on $K_0$ . It is easy to see that they correspond to nonconvex gluings along circles of radius larger than $L$ . Hence, until the integral $R$ takes the value such that the outer boundary component of $W$ , which has radius $R+L$ , attains this gluing circle, there can be no motion in a neighbourhood of this gluing edge. At the instant when $R$ attains this value, a critical circle appears, which thickens further to a domain of possible motion overlapping with two elementary billiard tables. In the molecule this also corresponds to an atom $A$ (see Figure 24, c).

The third type of critical points on the polygonal curve $K$ is minimum points that originally lie on $K_0$ under the line $y=L$ . They correspond to nonconvex gluings of billiard tables such that the radius of the gluing circles of these annuli is less that the radius of Larmor circles. For greater values of $R$ the fact that the line $y=L-R$ passes through such a point corresponds to the merger of two annuli lying on two sheets in the domain of possible motion, along their common nonconvex circle (see Figure 25, a).

The fourth type of critical points on the graph $K$ are minima obtained by the mirror reflection of maxima lying on $K_0$ . These maxima correspond to convex gluings of sheets. When the outer circle of the domain $W$ attains the circular edge of this convex gluing, a bifurcation occurs in which the domains of possible motion merge. This corresponds to the line $y=R+L$ passing through a point of the above type (see Figure 25, b).

Figure 25.Transformations of domains of possible motion: (a) two billiard tables (shown locally) glued along the nonconvex (inner) boundary; (b) gluing two tables along the convex (outer) boundary.

Extrema on the graph $M$ are also of two types, minima and maxima. Minima correspond to convex gluings of billiard tables. When $R$ attains the corresponding value, the inner boundary of $W$ attains the gluing edge. As $R$ grows further, the domain $W$ becomes disjoint from the corresponding billiard tables, so there can be no motion. Thus we obtain an atom $A$ see Figure 26, a).

Maxima on the graph $M$ correspond to nonconvex gluings of billiard tables. The inner boundary of the domain $W$ attains this gluing edge, and the domain of possible motion tears apart along the gluing circle (Figure 26, b).

Figure 26.Transformations of domains of possible motion: (a) two billiard tables (shown locally) glued along the nonconvex (inner) part of the boundary; (b) gluing two tables along the convex (outer) part of the boundary.

We have shown how the domain of possible motion changes in a neighbourhood of the image of each particular critical cycle. To calculate 3-atoms we must consider such changes in combination. As shown already, free vertices in the graph obtained correspond to atoms $A$ . In the other cases there are two possibilities: on the corresponding level $y=\mathrm{const}$ we have $k$ extrema of the third type and $l$ ones of the fourth type which lie on the curve $K$ (this is possible for $R<L$ ); or on this level $y=\mathrm{const}$ we have $k$ extrema of the fourth type on $K$ and $l$ maxima on $M$ (this is possible for $R>L$ ). Note that in either case the critical leaf of the required saddle atom is homeomorphic to the critical leaf of $B_{k+l}$ . Indeed, let us partition the relevant domain of possible motion into radial line segments. Two velocity vectors correspond to each point on such a segment (see Lemma 3), except for the two points on the boundary of the domain of possible motion and the $k+l$ points corresponding to critical circles. In the first case both bifurcations are, as shown above, the mergers of domains of possible motion. Hence this corresponds to a 3-atom $B_{k+l}$ in the isoenergy manifold $Q^3$ . In the second case $k$ critical circles correspond to gluing domains of possible motion and $l$ circles correspond to tearing apart such domains. Taking account of the relative arrangement of these gluings and tearings, it is easy to see that in the isoenergy manifold $Q^3$ such a bifurcation corresponds to a 3-atom $B_\tau$ (as shown graphically in Figure 23, c, d).

Now the proof is complete.

Now, to one or two free boundary components of the topological billiard $bC$ we glue a billiard table bounded by the circle of appropriate radius. The configuration space of such a billiard is homeomorphic to a 2-disc $D^2$ (the billiard $bD$ ) or a 2-sphere $S^2$ (the billiard $bS$ ), respectively. Note that the algorithm calculating rough molecules is quite as before. In fact, the only difference of $bD$ and $bS$ from the billiard $bC$ is that now there can be domains of possible motion containing one or two centres of circular billiard tables (so that these domains are homeomorphic to $D^2$ or $S^2$ , respectively). As shown before, the preimages of such domains of possible motion are also homeomorphic to 2-tori. Hence the level $y=0$ in Figure 23, a, is not itself critical, and therefore the rough molecule shown in Figure 23, b, remains the same.

The algorithm constructing a rough molecule outputs immediately the type of its structure. Namely, we have the following result.

Proposition 4. The following assertions hold.

The Fomenko invariant of the billiards $bC$ , $bD$ and $bS$ is a tree.

For sufficiently large values of the energy $L$ this tree has the symmetric form $W$ – $W$ , where $W$ is a rooted subtree, and the two symmetric halves of the molecule are joined by an edge incident to both roots.

Proof. We prove the first part by contradiction. In fact, if the graph contains a cycle, then there is a noncontractible circle in the domain bounded by

$K$ and

$M$ that is mapped to this cycle when we take the quotient of the domain (by contracting the horizontal segments). However, the domain between these two polygonal curves is always simply connected. This is a contradiction.

To prove the second part note that for large values of $L$ the straight line $y=L$ lies above the polygonal curve $K_0$ . Hence $K$ and $K_0$ coincide, and the domain bounded by $K$ and $M$ is symmetric relative to the axis $y=0$ . Consequently, the tree is symmetric. Now note that only extrema of the first and third type positioned above this axis can lie on $K$ . In other words, one of the two symmetric halves of the tree (which we denote by $W$ ) contains only atoms $A$ and atoms $B_k$ uniting tori, which means that we have a rooted tree. Finally, note that the level $y=0$ corresponds to a unique horizontal segment in the domain between the polygonal curves. Hence the two symmetric rooted subtrees are connected by a single edge.

The proof is complete.

Now consider a topological billiard $bC$ and assume that an even number of elementary tables are glued together, and the free boundary components have the same radius. Next we glue these free boundary components so as to make the configuration space homeomorphic to the 2-torus $T^2$ (we leave out the case when the configuration space is homeomorphism to a Klein bottle). We denote the resulting topological billiard by $bT$ .

Definition 4. Let $\tau$ be a finite sequence of positive integers defined up to a cyclic permutation of its elements. Let $n=\sum{\tau_i}>1$ . Let $\widehat{C}_\tau$ be the saddle atom defined as follows: we cut the 2-atom $C_n$ transversally into $|\tau|$ connected pieces so that for each $i$ the $i$ th piece contains precisely $\tau_i$ critical points. Then we glue these pieces back with twisting along the same cuts (Figure 27). Multiplying this 2-atom by a circle we obtain a 3-atom. This 3-atom bifurcates $k=\sum_{2\mid i}\tau_i$ Liouville tori into $l=\sum_{2\nmid i}\tau_i$ ones.

Figure 27.The 2-atom $\widehat{C}_\tau$ for $\tau=(2,2,1,1)$ .

Although the constructions of $B_\tau$ and $\widehat{C}_\tau$ are very similar, we cannot always embed the 2-atom $\widehat{C}_\tau$ in the plane. The simplest example is $\widehat{C}_{1,1}$ , which is better known as an atom $C_1$ (see [4]).

The algorithm for calculating the rough molecule for $bT$ is similar to the one for $bC$ . Namely, fix a value of the first integral $L$ . Consider the polygonal curves $K_0$ , $K$ and $M$ in the $0xy$ -plane constructed similarly to the previous case and map this plane onto a cylinder with generator $0y$ so that the free vertices of each graph are taken to the same point (Figure 28, a). In other words, $K_0$ and $M$ make up the profile of a billiard as before. We partition the domain between the curves $K$ and $M$ into horizontal layers and contract each layer to a point in the ambient space $\mathbb{R}^3$ of the cylinder. In the graph obtained we add a simple edge to each edge corresponding to a noncontractible layer on the cylinder and assign the indices $A$ to free vertices. If a vertex of the graph corresponds to a contractible layer on the cylinder, then we assign an index $B_\tau$ to it as previously. If it corresponds to a noncontractible layer, then we assign an index $\widehat{C}_\tau$ to it in accordance with the same rule (Figure 28, b).

Figure 28.The construction of a rough molecule.

Lemma 5. The graph with indexed vertices constructed above (see Figure 28, b) is a rough molecule of the topological billiard $bT$ for the fixed energy level $L$ .

Proof. In the case when the level set

$y=\mathrm{const}$ is contractible on the cylinder we can prove this lemma similarly to Lemma 4. Assume that the graphs

$K$ and

$M$ contain no minima or maxima on the level

$y=\mathrm{const}$ and this level is noncontractible in the domain between these two curves. Then the domain of possible motion coincides with the whole of the billiard table

$bT$ , which is homeomorphic to the torus. A trajectory of the point mass forms a winding on this torus, and the point moves in one of the two directions on its meridian. Hence there are two Liouville tori in the preimage of such a domain of possible motion, which explains why multiple edges must be added.

Now let the level $y=\mathrm{const}$ be noncontractible again, but let some extrema of the graphs $K$ and $M$ lie on it. Note that the preimage of the corresponding domain of possible motion is homeomorphic to the critical leaf of the 3-atom $C_n$ . Indeed, let us partition this domain of possible motion into radial segments. The resulting cross-sections of $bT$ are homeomorphic to circles. Two velocity vectors correspond to each point in such a circle, with the exception of the $n$ points corresponding to critical circles, in which the two circles in the preimage intersect.

Here, similarly to the billiard $bC$ , we have two possibilities: either $k$ extrema of the third type and $l$ extrema of the fourth type on the curve $K$ lie on this level (this is possible for $R<L$ ); or $k$ extrema of the fourth type on $K$ and $l$ maxima on $M$ lie on it (this is possible for $R>L$ ). In the first case both types of bifurcations are, as shown above, the mergers of domains of possible motion into a single domain, which is homeomorphic to a torus. Hence this corresponds to a 3-atom $C_{k+l}$ in the isoenergy manifold $Q^3$ . In the second case $k$ critical circles correspond to gluing together domains of possible motion and $l$ critical circles corresponds to tearing them apart. Taking the mutual position of these gluings and tearings on the critical level into account, we can see that such a bifurcation in the isoenergy manifold $Q^3$ corresponds to a 3-atom $\widehat{C}_\tau$ . Now the proof is complete.

Note that if there is an atom $\widehat{C}_\tau$ with $|\tau|>1$ in the molecule, then it is the unique atom of this form there (all other levels $y=\mathrm{const}$ are contractible on the cylinder). Thus we have described all integrable topological magnetic billiard with orientable configuration space from the standpoint of rough Liouville equivalence.

Remark 2. Note that even for a topological billiard the rough molecule corresponding to the isointegral manifold $Q_R^3$ for a fixed value $R=R_0$ coincides with the rough molecule corresponding to the isoenergy manifold $Q^3$ for the fixed value of energy $L_0=R_0$ . Indeed, as we noted above, the domain of possible motion for fixed values of $L$ and $R$ is the intersection of the domain $W$ bounded by the circles of radii $R+L$ and $|R-L|$ , with the billiard table. Thus the domain of possible motion for the values $L_1$ and $R_1$ of the first integrals coincides with the domain for the values $L_2=R_1$ and $R_2=L_1$ . Hence the bifurcations of domains of possible motion are also the same and occur in the same order, which means that the rough molecules coincide.

Proposition 5. The bifurcation diagram of a magnetic topological billiard consists of segments of the lines $R+L=R_j$ and rays of the lines $|R - L| = R_j$ in the first coordinate sector, where the $R_j$ are the radii of gluing circles, and also of segments of the lines $R+L=r_i$ , where the $r_i$ are the radii of free inner boundary circles and the rays $|R-L|=R_i$ , where the $R_i$ are the radii of free outer boundary circles (Figure 29).

Figure 29.The bifurcation diagram of a magnetic topological billiard.

Proof. As shown before, bifurcations occur when either the outer circle of radius

$R+L$ of the domain

$W$ or its inner boundary circle of radius

$|R-L|$ attains the gluing circle of radius

$R_i$ or the free boundary component of the billiard table of radius

$r_i$ .

Now we calculate the marks corresponding to the rough molecules we have constructed. Note that the images on the billiard table of critical circles of each atom coincide with gluing circles of elementary billiard tables (sheets). We orient them in the direction of the velocity of the point mass. Note that there are critical circles with opposite orientations on atoms $B_\tau$ and $\widehat{C}_\tau$ for $|\tau|>1$ (Figure 30, g). Thus, in what follows we consider only molecules without such atoms. We divide the other 3-atoms into two groups, 1) a minimal atom $A$ , an atom $B_k$ bifurcating (gluing) $k+1$ Liouville tori into a single torus, and an atom $C_k$ bifurcating (gluing) $k$ Liouville tori into two tori (Figure 30, a, b, c, d); 2) a maximal atom $A$ , an atom $B_k$ bifurcating (cutting) a Liouville torus into $k+1$ tori and an atom $C_k$ bifurcating (cutting) two Liouville tori into $k$ tori (Figure 30, e, f). Note that on all atoms in the first group critical circles are oriented anticlockwise, and on atoms in the second they are oriented clockwise.

Figure 30.The orientations of critical circles: (a), (b) a Larmor circle and the critical circle of a minimal 3-atom $A$ ; (c), (d), (e) the part of the domain of possible motion that lies on an elementary billiard table, the critical circle of a saddle 3-atom and a trajectory tangent to it; (f) a Larmor circle and the critical circle of a maximal 3-atom $A$ ; (g) two critical trajectories of a 3-atom $B_\tau$ or $\widehat{C}_\tau$ with opposite orientation.

Theorem 2. The Fomenko-Zieschang invariant of a magnetic topological billiard has the following form:

the rough molecule is produced by the algorithm presented in Lemma 4 for billiards $bC$ , $bD$ and $bS$ , and in Lemma 5 for a billiard $bT$ ;

the mark $r$ is zero on all edges incident to atoms $A$ and is infinite on all other edges (so that a unique family exists);

the mark $\varepsilon$ is equal to $+1$ on all edges connecting two atoms in the same group or two atoms $A$ , and to $-1$ otherwise;

the mark $n$ of the unique family is zero for billiards $bC$ and $bT$ , $\pm1$ for a billiard $bD$ and $\pm2$ for $bS$ .

Proof. We start with topological billiards

$bC$ and

$bT$ . As noted before, the images of critical circles of any atom are gluing circles of the topological billiard. For a saddle atom they are homologous to the cycle

$\lambda$ in an admissible basis (see Figure 30, c, d). We select the complementary cycle

$\mu$ so that its image on the billiard table lies on a single radius. Note that once we have fixed a radius, the preimage of the points on it corresponds to horizontal segments between the curves

$K$ and

$M$ on the plane or cylinder (see the construction of the rough molecule). In fact, each segment contractible to a point corresponds to a boundary circle of some two-dimensional atom. Thus, first, all cycles we have selected are indeed transversal cross-sections of some regular leaves in the corresponding three-dimensional atoms and, second, for each 3-atom selected they are the boundary circles of the same two-dimensional cross-section. Since the pair of cycles

$\lambda$ ,

$\mu$ of a saddle atom is homologous to the pair

$\lambda$ ,

$\mu$ of another saddle atom, the gluing matrix on an edge connecting two saddle atoms has the form

$\begin{pmatrix}\pm1 & 0\\0 & \mp1\end{pmatrix}$ . Hence

$r=\infty$ , and the contribution to the mark

$n$ from such edges is zero. Furthermore, as mentioned before, if these saddle atoms belong to the same group, then their cycles

$\lambda$ are similarly oriented. Hence

$\varepsilon=1$ . Otherwise

${\varepsilon=-1}$ .

By contrast, on atoms $A$ the cycle $\mu$ is homologous to the critical circle (see Figure 30, a, b), and the image of the vanishing cycle $\lambda$ lies on a single selected radius. Hence on an edge connecting an atom $A$ and a saddle atom the gluing matrix has the form $\begin{pmatrix}0 &\pm1\\ \pm1 & 0 \end{pmatrix}$ . Therefore, $r=0$ on edges, and the contribution to the mark $n$ from such edges is also zero. As in the previous case, the mark $\varepsilon$ is $1$ if the saddle atom and the atom $A$ belong to the same group, and $\varepsilon=-1$ otherwise. The case when an edge connects two atoms $A$ was considered above for an annulus-like billiard.

Now consider a topological billiard $bD$ . As noted before, it is different from $bC$ by the unique domain of possible motion which appears for $R=L$ and contains the centre of the circular billiard table. We find the unique edge $e$ in the rough molecule on which the point mass corresponding to this domain of possible motion can lie. The gluing matrices corresponding to all other edges are analogous to the case of $bC$ . If an edge $e$ connects two saddle atoms, then, as in the case of $bC$ , the cycles $\lambda$ on these atoms coincide up to orientation. To find the relation between the cycles $\mu$ , we use the cycle $\gamma$ similar to the one for the usual circular billiard (see Figure 12). Thus the gluing matrix on $e$ has the form $\begin{pmatrix}\pm1 & 0\\(-1)^i & \mp1\end{pmatrix}$ , where $i\in{0,1}$ . Hence $r=\infty$ and the contribution to $n$ is $\pm1$ (depending on the orientation of $Q^3$ , which affects the value of $i$ ). As before, $\varepsilon=1$ if the saddle atoms belong to the same group and $\varepsilon=-1$ otherwise.

On the other hand, if $e$ connects a saddle atom with $A$ , then using the same cycle $\gamma$ we can show that the gluing matrix has the form $\begin{pmatrix}0 & \pm1\\ \pm1 &(-1)^i\end{pmatrix}$ . Hence $r=0$ , and the contribution to the mark $n$ is $\pm1$ again. As before, $\varepsilon=1$ if the saddle atom and atom $A$ belong to the same group, and $\varepsilon=-1$ otherwise. The case when $e$ connects two atoms $A$ was considered in the case of a circular billiard.

Finally, we look at a billiard $bS$ . Consider the domain of possible motion corresponding to $R=L$ . Then either this domain has two connected components, each containing the centre of a circular table, or this domain consists of a unique connected component homeomorphic to a sphere, and then it contains both centres.

In the first case the rough molecule contains two edges analogous to $e$ for a billiard $bD$ . Hence the marks $r$ and $\varepsilon$ on such edges coincide with the marks on the edge $e$ . Each of these edges contributes $\pm1$ to the mark $n$ We can show that for a suitable choice of the orientation of $Q^3$ all contributions to $n$ are positive, so both contributions must have the same sign.

In the second case the auxiliary cycle $\gamma$ has a slightly different structure. Namely, its image on the cylinder consists of two parts, which coincide with the images of the old cycle $\gamma$ and lie on two circular billiard tables. Thus, the gluing matrices have the form $\begin{pmatrix}\pm1 & 0\\2(-1)^i &\mp1\end{pmatrix}$ for an edge connecting saddle atoms and $\begin{pmatrix} 0 &\pm1\\ \pm1 &2(-1)^i\end{pmatrix}$ for an edge connecting a saddle atom and an atom $A$ . The marks $r$ and $\varepsilon$ are analogous to the marks for $bD$ , and the contribution of this edge to $n$ is $\pm2$ .

The proof is complete.

§ 9. A magnetic topological billiard is Liouville equivalent to a geodesic flow on a surface of revolution in a potential field

We show in conclusion which known dynamical systems with two degrees of freedom are Liouville equivalent to topological magnetic billiards. Consider an integrable geodesic flow on a surface of revolution homeomorphic to a sphere, in a potential field. Integrable systems of such type were investigated by Kantonistova [13] from the standpoint of the topology of the Liouville foliations. More precisely, such a geodesic flow on a manifold $\mathcal{M}\simeq S^2$ is defined by a pair of functions of one variable $(f(r), V(r))$ , where $f\colon [0, I]\to\mathbb{R}$ defines the metric of revolution $ds^2=dr^2+f^2(r)d\phi^2$ on $\mathcal{M}$ , and $V$ is a potential independent of the angle variable $\phi$ . The Hamiltonian of this system is

$\begin{equation*} H=\frac{p_r^2}{2}+\frac{p_\phi^2}{2f^2(r)}+V(r), \end{equation*} \notag$

and the additional first integral

$F=p_\phi$ is linear in momenta. By [13] the following result holds.

Theorem 3 (Kantonistova). Consider the geodesic flow on a surface of revolution in a potential field which is defined by a pair of functions $(f, V)$ . Let $Q\subseteq Q_h^3=\{H=h\}$ be the connected component of a nonsingular isoenergy surface, on which the integral $K$ is a Bott function. Then the following assertions hold.

The Fomenko-Zieschang invariant of $Q$ is symmetric relative to the $h$ -axis, so that it has the form $W$ – $W$ . In addition, $W$ is a rooted tree with free vertices corresponding to 3-atoms $A$ and other vertices corresponding to 3-atoms $B_k$ for arbitrary $k$ .

The marks on edges of type $A$ – $B_k$ of the molecule are $r=0$ and $\varepsilon=1$ .

The marks on edges of type $B_l$ – $B_k$ , where both saddle atoms lie on one of the two symmetric halves of the molecule $W$ – $W$ are $r=\infty$ and $\varepsilon=1$ .

The marks on an edge of type $B_k$ – $B_k$ connecting the roots of the symmetric halves of the molecule are $r=\infty$ and $\varepsilon=-1$ .

The mark $n$ on the unique family is $0$ , $1$ or $2$ depending on the number of points of rank 0 in the submanifold $\{H<h\}$ .

Now we show that for some values of the first integral $L$ a magnetic topological billiard is Liouville equivalent to the geodesic flow on a surface of revolution in a potential field. Namely, the following theorem holds.

Theorem 4. Consider the geodesic flow on a surface of revolution in a potential field which is defined by a pair of functions $(f, V)$ . Fix a nonsingular value of the energy of this flow such that the assumptions of Theorem 3 are fulfilled. Then there exists a magnetic topological billiard whose Fomenko-Zieschang invariant coincides for large values of the first integral $L$ with the invariant corresponding to the fixed value of the energy of the geodesic flow. Hence these two systems are Liouville equivalent.

Proof. Assume that a Fomenko-Zieschang invariant

$W$ –

$W$ corresponds to the fixed value of the energy of the geodesic flow. Fix this invariant. We shall have proved the theorem once we have found a billiard in the class

$bC$ ,

$bD$ or

$bS$ (depending on the value of the mark

$n$ ) such that for sufficiently large values of

$L$ its Fomenko-Zieschang invariant coincides with the fixed rough molecule. As concerns the marks on this molecule, they will automatically coincide with the ones on the fixed rough molecule in accordance with Theorems 2 and 3 and Proposition 4.

To construct the required billiard we present an inverse algorithm to the one constructing a rough molecule for a billiard system (see Figure 31).

Figure 31.The algorithm recovering a billiard from a molecule: (a) the rooted subgraph $W$ and its partition into the groups $f_i$ ; (b) placing the vertices of the polygonal curve $K$ in accordance with the algorithm; (c) the resulting cross-section of the required topological billiard.

Step 0. Consider the subgraph $W$ . Let its height be $N-1$ . Divide its atoms into groups in accordance with their distances to the root. Let $f_i$ denote these groups, where $i$ is the distance. Note that $f_0$ contains a unique atom $B_k$ (to which we assign the index 1).

Step 1. In the half-plane $\{0xy\colon x>0\}$ , on the line $y=1$ we choose $k$ points with coordinates $x_1^1<x^2_1<\dots<x^k_1$ , where $k$ is the complexity of the atom in the group $f=f_0$ .

Step 2. We number the atoms in $f_1$ by pairs of integers $(1,1)$ , $(1,2)$ , …, $(1,N_2)$ , where $N_2$ is the number of atoms, and the first integer in each pair indicates that these atoms are adjacent to atom no. 1. Let the atom $(1,j)$ have complexity $k_j$ . Then choose $k_j$ points with coordinates $x_1^{j-1}<x^1_{1,j}<\dots<x^{k_j}_{1,j}<x^j_1$ on the line $y=2$ . Next we describe the steps with numbers $3\leqslant i\leqslant{N}$ .

Step $i$ . Assume that the group $f_{i-1}$ contains $N_i$ atoms. Number them by tuples of $i$ integers $(1,j_2,j_3,\dots,j_i)$ , where $1\,{\leqslant}\, j_i\,{\leqslant}\, N_i$ . Here atom no. $(1,j_2,j_3,\dots,j_{i-1}, j_i)$ is adjacent to atom no. $(1,j_2,j_3,\dots,j_{i-1})$ in the group $f_{i-2}$ (recall that $W$ is a rooted tree). Assume that this atom has complexity $k_{j_i}$ . Then choose $k_{j_i}$ points with coordinates $x_{1,j_2,j_3,\dots,j_{i-1}}^{j_i-1}<x^1_{1,j_2,j_3,\dots,j_{i-1}, j_i}<\dots<x^{k_{j_i}}_{1,j_2,j_3,\dots,j_{i-1}, j_i}<x_{1,j_2,j_3,\dots,j_{i-1}}^{j_i}$ on the line $y=i$ . Here we assume that $x_{1,j_2,j_3,\dots,j_{i-1}}^{0}=x_{1,j_2,j_3,\dots,j_{i-2}}^{j_{i-1}-1}$ and $x_{1,j_2,j_3,\dots,j_{i-1}}^{k_{j_{i-1}+1}}=x_{1,j_2,j_3,\dots,j_{i-2}}^{j_{i-1}}$ , where $k_{j_{i-1}}$ is the complexity of atom no. $(1,j_2, j_3,\dots,j_{i-1})$ .

Step $N+1$ . Connect the points constructed at the previous steps by a polygonal curve $K$ in the increasing order of their coordinates $x$ . Mirror reflect this curve in the axis $y=0$ (let $M$ denote the resulting curve). In combination $K$ and $M$ make up the profile of a topological billiard in the class $bC$ . Recover the original billiard from this profile.

Step $N+2$ . If the mark $n$ on the family in the original molecule is equal to 1, then glue an elementary circular table (sheet) to a free boundary component of the billiard constructed. If the mark $n$ is equal to 2, then glue such circular sheets to both free boundary components.

The billiard constructed has the rough molecule we have fixed for $L=N+1$ . In fact, consider the group $f_i$ of atoms. If it contains an atom $A$ with number $(1,j_2,j_3,\dots,j_{i-1}, j_i)$ , then this atom is not adjacent to any atoms in the preceding groups. Hence the vertices of the curve $K$ closest to the point $x^{1}_{1,j_2,j_3,\dots,j_{i-1}, j_i}$ lie below the line $y=i$ , that is, this is a maximum point on $K$ . We can show in a similar way that points on the polygonal curve which correspond to saddle atoms are local minima. In addition, all local minima corresponding to the same atom $B_k$ lie on a contractible horizontal segment of the line $y=i$ (for otherwise this atom is adjacent to two atoms in a preceding group), and only these minima lie there (for otherwise there exists another atom in the same group $f_i$ which is adjacent to $B_k$ ). Hence the above algorithm is indeed inverse to the one in Lemma 4.

The proof is complete.

Remark 3. This theorem means that the class of Liouville foliations of nonsingular isoenergy manifolds for geodesic flows on surfaces of revolution with potential lies fully (up to Liouville equivalence) in the analogous class of magnetic topological billiards. Moreover, the class of billiards is wider because rough molecules corresponding to it are not necessarily symmetric (see Figure 23).



Bibliography

1.	V. V. Fokicheva (Vedyushkina), “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics”, Mat. Sb., 206:10 (2015), 127–176 ; English transl. in Sb. Math., 206:10 (2015), 1463–1507
2.	M. Bialy and A. E. Mironov, “Algebraic non-integrability of magnetic billiards”, J. Phys. A, 49:45 (2016), 455101, 18 pp.
3.	V. V. Vedyushkina, “The Fomenko-Zieschang invariants of nonconvex topological billiards”, Mat. Sb., 210:3 (2019), 17–74 ; English transl. in Sb. Math., 210:3 (2019), 310–363
4.	A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, v. 1, 2, Udmurtian University Publishing House, Izhevsk, 1999, 444 pp., 447 pp. ; English transl., Chapman & Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp.
5.	E. A. Kudryavtseva and A. A. Oshemkov, “Bifurcations of integrable mechanical systems with magnetic field on surfaces of revolution”, Chebyshevskii Sb., 21:2 (2020), 244–265 (Russian)
6.	A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Uspekhi Mat. Nauk, 44:1(265) (1989), 145–173 ; English transl. in Russian Math. Surveys, 44:1 (1989), 181–219
7.	A. T. Fomenko and H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Izv. Akad. Nauk SSSR Ser. Mat., 54:3 (1990), 546–575 ; English transl. in Math. USSR-Izv., 36:3 (1991), 567–596
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12.	V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Ross. Akad. Nauk Ser. Mat., 81:4 (2017), 20–67 ; English transl. in Izv. Math., 81:4 (2017), 688–733
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Citation: V. V. Vedyushkina, S. E. Pustovoitov, “Classification of Liouville foliations of integrable topological billiards in magnetic fields”, Sb. Math., 214:2 (2023), 166–196

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\paper Classification of Liouville foliations of integrable topological billiards in magnetic fields

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\pages 166--196

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This publication is cited in the following 3 articles:

G. V. Belozerov, A. T. Fomenko, “Orbital invariants of billiards and linearly integrable geodesic flows”, Sb. Math., 215:5 (2024), 573–611
S. E. Pustovoitov, “Issledovanie struktury sloeniya Liuvillya integriruemogo ellipticheskogo billiarda s polinomialnym potentsialom”, Chebyshevskii sb., 25:1 (2024), 62–102
A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrable systems”, Russian Math. Surveys, 78:5 (2023), 881–954

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§ 1. Introduction

§ 2. Requisite definitions

§ 3. Statement of the problem. Trajectories

§ 4. Regions of possible motion. Bifurcation diagrams

§ 5. Fomenko-Zieschang invariants. Isoenergy manifolds

§ 6. Liouville equivalent systems

§ 7. Isointegral manifold Q3RQ^3_R

§ 8. Topological billiards

§ 9. A magnetic topological billiard is Liouville equivalent to a geodesic flow on a surface of revolution in a potential field

Bibliography

§ 7. Isointegral manifold $Q^3_R$