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In the theory of integrable Hamiltonian systems, an important role is played by the study of Liouville foliations and bifurcations of their leaves. In the compact case, the problem is solved, but the noncompact case remains mostly unknown. The main goal of this article is to formulate the noncompact problem and to present a set of examples of Hamiltonian systems, giving rise to noncompact bifurcations and Liouville leaves.

In the paper, eight classes of integrable billiards are studied; in particular, classes introduced by the authors: elementary, topological, billiard books, billiards on the Minkowski plane, geodesic billiards on quadrics in three-dimensional Euclidean space, billiards in a magnetic field, and also a class containing all of the ones above. It turns out that, in the class of billiard books, topological obstacles to implementation occurred, for example, for the “twisted” Lagrange top (as we conventionally call a modification of the usual Lagrange top which we had discovered) for one of energy zones. We indicate this obstacle explicitly. It turns out further that this system can still be implemented in the class of magnetic billiards.

Description of bifurcations and symmetries of integrable systems is an important branch of geometry that has many applications. Important results have been obtained recently in the descriptions of bifurcations of integrable billiards and in modelling of Hamiltonian systems of mechanics and dynamics by billiards. The paper contains interesting problems, as well as a research program for the near future. In the closing of the paper, the results allowing one to describe hidden symmetries of Hamiltonian bifurcations are given as an example of a work close to billiards subject.

In the theory of integrable Hamiltonian systems with two degrees of freedom, widely known are integrable systems having integrals of high degrees, namely, 3 and 4. Examples are the Kovalevskaya system and its generalizations—the Kovalevskaya–Yehia system and the Kovalevskaya system on the Lie algebra so(4) the Goryachev–Chaplygin–Sretensky, Sokolov, and Dullin–Matveev systems. It is shown that, at a number of isoenergy 3-surfaces, the third and fourth degrees of integrals of these systems can be reduced by using integrable billiards bounded by arcs of confocal quadrics. Moreover, the integrals of degree 3 and 4 are reduced to the same canonical quadratic integral on a billiard.

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.

Saddle or hyperbolic singularities of Liouville foliations of integrable Hamiltonian systems are discussed. We observe new and classical results on their classification, representation and invariants with respect to topological equivalence depending on number of degrees of freedom. Then criterion of their component-wise stability by A.A. Oshemkov and its application are reminded. At last, we discuss saddle singularities of famous dynamical and physical systems, particularly problem of realization (modeling) of Liouville foliations and their singularities (A.T. Fomenko billiard conjecture) by integrable billiards. New result is obtained: loop molecules of all saddle-saddle singularities with one equilibrium are modeled by billiard books, i.e. integrable billiards on CW-complexes introduced by V.V. Vedyushkina.

We consider several topological integrable billiards and prove that they are Liouville equivalent to many systems of rigid body dynamics. The proof uses the Fomenko– Zieschang theory of invariants of integrable systems. We study billiards bounded by arcs of confocal quadrics and their generalizations, generalized billiards, where the motion occurs on a locally planar surface obtained by gluing several planar domains isometrically along their boundaries, which are arcs of confocal quadrics. We describe two new classes of integrable billiards bounded by arcs of confocal quadrics, namely, non- compact billiards and generalized billiards obtained by gluing planar billiards along non-convex parts of their boundaries. We completely classify non- compact billiards bounded by arcs of confocal quadrics and study their topology using the Fomenko invariants that describe the bifurcations of singular leaves of the additional integral. We study the topology of isoenergy surfaces for some non-convex generalized billiards. It turns out that they possess exotic Liouville foliations: the integral trajectories of the billiard that lie on some singular leaves admit no continuous extension. Such billiards appear to be leafwise equivalent to billiards bounded by arcs of confocal quadrics in the Minkowski metric.

It is well known that surgeries of closures of solutions for integrable nondegenerate Hamiltonian systems with two degrees of freedom at a level of constant energy are classified by the so-called 3-atoms. These surgeries correspond to singular leaves of the Liouville foliation of three-dimensional isoenergetic surfaces. In this paper we prove the Fomenko conjecture that all such surgeries are modeled by integrable topological two-dimensional billiards (billiard books).

This paper is algebraic and topology study of the manifolds of constant energy of integrable Hamiltonian systems with two degrees of freedom. The Liouville foliation defines the topology of isoenergy manifold, but on any isoenergy manifold there are many non-equivalent Hamiltonian systems. We give some review of recent papers on homotopy invariants and their relation with Fomenko-Zieschang invariants. Also, we discuss relatively new results about Reidemeister torsion and applications in the theory of Hamiltonian systems. The last section is efficiency demonstration of Fomenko-Zieschang invariants in concrete mechanic system. Let us note that many known Hamiltonian systems have been investigated in terms of Fomenko-Zieschang invariants.

The main goal of this paper is to demonstrate how the theory of invariants for integrable Hamiltonian systems with two degrees of freedom created by A.T. Fomenko, H. Zieschang, and A.V. Bolsinov helps to establish Liouville and orbital equivalence of some classical integrable systems. Three such systems are treated in the article: the Euler case in rigid body dynamics, the Jacobi problem about geodesics on the ellipsoid and the Chaplygin case in dynamics of a rigid body in fluid. The first two systems were known to be Liouville and even topologically orbitally equivalent (Fomenko, Bolsinov). Now we show that the Chaplygin system is orbitally equivalent to the Euler and Jacobi systems.

This work provides an outline of several results concerning topology, Lie algebra, orbits and dynamics of some integrable systems on them. All the results in this paper were obtained by the authors and the participants of the Seminar on Modern Geometry and Its Applications held in the Faculty of Mechanics and Mathematics of the Moscow State University, organized by A. T. Fomenko.

Maps are studied, i.e., cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms which encode bifurcations of Liouville fibrations of non-degenerate integrable Hamiltonian systems. Any finite group G is proved to be a symmetry group of an orientable map (of an atom). Moreover, one such map X(G) is constructed algorithmically. Upper bounds are obtained for the minimal genus Mg(G) of an orientable map with the given symmetry group G and for the minimal number of vertices, edges, and sides of such maps.

K-theory emerged as an independent part of topology in the late 1950s, when the limits of the possibilities of the methods based on spectral sequences and cohomology operations (and studied in Chaps. III–V) became visible. Progress in homotopy topology considerably slowed down, the leading topologists got involved in cumbersome calculations, the results became less and less impressive, and to obtain them one had to combine a great inventiveness with a readiness to do a huge amount of tedious work. To get these activities revived a radical method was needed. And it was found! It consisted of replacing cohomology as the basic homotopy invariant by an entirely new object, the so-called K-functor.

Description of the rigid body dynamics is a complex problem, which goes back to Euler and Lagrange. These systems are described in the six-dimensional phase space and have two integrals the energy integral and the momentum integral. Of particular interest are the cases of rigid body dynamics, where there exists the additional integral, and where the Liouville integrability can be established. Because many of such a systems are difficult to describe, the next step in their analysis is the calculation of invariants for integrable systems, namely, the so called Fomenko–Zieschang molecules, which allow us to describe such a systems in the simple terms, and also allow us to set the Liouville equivalence between different integrable systems. Billiard systems describe the motion of the material point on a plane domain, bounded by a closed curve. The phase space is the four-dimensional manifold. Billiard systems can be integrable for a suitable choice of the boundary, for example, when the boundary consists of the arcs of the confocal ellipses, hyperbolas and parabolas. Since such a billiard systems are Liouville integrable, they are classified by the Fomenko–Zieschang invariants. In this article, we simulate many cases of motion of a rigid body in 3-space by more simple billiard systems. Namely, we set the Liouville equivalence between different systems by comparing the Fomenko–Zieschang invariants for the rigid body dynamics and for the billiard systems. For example, the Euler case can be simulated by the billiards for all values of energy integral. For many values of energy, such billard simulation is done for the systems of the Lagrange top and Kovalevskaya top, then for the Zhukovskii gyrostat, for the systems by Goryachev–Chaplygin–Sretenskii, Clebsch, Sokolov, as well as expanding the classical Kovalevskaya top Kovalevskaya–Yahia case.

Lecture 12 Main Definitions and Constructions Besides the homotopy groups πn(X), there are other series of groups coresponding in a homotopy invariant way to a topological space X; the most notable are homology and cohomology groups, H n (X) and Hn (X).

As we demonstrated in the previous lecture, the information about the action of stable cohomology operations may be used for computing stable homotopy groups. If we know the cohomology of the space X, we can find, relatively easily, the “stable part” of the cohomology of the first killing space of X, and then the same for the second, the third, and so on killing spaces. Hurewicz’s theorem gives us, every time, the corresponding homotopy group. This method (the Serre method) does not permit us, however, to find the homotopy groups without dealing with other difficulties.

Let n, q be two integers and let π, G be two Abelian groups. We say that a cohomology operation\(\varphi\)of type (n, q, π, G) is given if for every CW complex X a map \(\varphi _{X}: H^{n}(X;\pi ) \rightarrow H^{q}(X;G)\) is given and is natural with respect to X, in the sense that the diagram

Let X and Y be topological spaces. Continuous maps f, g: X → Y are called homotopic (f ∼ g) if there exists a family of maps h t : X → Y, t ∈ I such that (1) \(h_{0} = f,h_{1} = g\); (2) the map H: X × I → Y, H(x, t) = h t (x), is continuous.

This article describes new results obtained in the theory of symmetries and singularities of integrable Hamiltonian systems, developed in recent years by the Fomenko school in Moscow State University. The Sadetovʼs proof of Mischenko–Fomenko conjecture, the correlation between the discriminant of the spectral curve and the bifurcation diagram and the theory of atoms for the two-dimensional integrable systems are discussed.

The first part of this work deals with the investigation and modeling of foliations generated by dynamic systems on their phase spaces and configuration spaces. In the second part, we speak in greater detail about curves and surfaces in three-dimensional space and their geometrical characteristics that can be useful for modeling polymer conformation.

Regular (maximally symmetric) cell decompositions of closed oriented 2-dimensional surfaces (that is, regular maps or regular abstract polyhedra) are considered. These objects are also known as maximally symmetric oriented atoms. An atom is reducible if it is a branched covering of another atom, with branching points at vertices of the decomposition and/or the centres of faces. The following two problems have arisen in the theory of integrable Hamiltonian systems: describe the irreducible maximally symmetric atoms; describe all the maximally symmetric atoms covering a fixed irreducible maximally symmetric atom. In this paper, these problems are solved in important cases. As applications, the following maximally symmetric atoms are listed: the atoms containing at most 30 edges; the atoms containing at most six faces; the atoms containing p or 2p, where p is a prime.Bibliography: 52 titles.

The paper is devoted to a short explanation of the topological classification (up to Liouville equivalence) of the integrable geodesic flows of two-dimensional surfaces of revolution with potential. The classification is given in the terms of so-called “marked molecules,” i.e., Fomenko–Zieschang invariants for integrable systems with two degrees of freedom on three-dimensional isoenergy sufraces.

Hamiltonian systems are considered to be the prime tool of classical and quantum mechanics. The proper investigation of such systems usually requires deep results from algebra and geometry. Here we present several results which in some sense go the opposite way: the knowledge about the integrable system enables us to obtain results on geometric and algebraic structures which naturally appear in such problems. All the results were obtained by employees of the Chair of Differential Geometry and Applications in Moscow State University in 2011–2012.

A method for calculating topological invariants of the foliation of a phase space into invariant Liouville tori in the case of integrable Hamiltonian systems with two degrees of freedom is put forward. The structure of this foliation is completely described for the Kovalevskaya integrable case in rigid body dynamics.

The present paper contains an exact topological classification of all nondegenerate Hamiltonian systems on smooth closed two-dimensional surfaces. Bibliography: 8 titles.

Physical and mechanical systems with four-dimensional phase space are considered. The classification of nondegenerate integral systems is studied. A “physical zone,’ i.e., the systems connected with real physical applications, is determined. Bibliography: 27 titles.

Classical and new results on integrable geodesic flows on two-dimensional surfaces are reviewed. The central question is the classification of such flows up to various equivalences, of which the following four kinds are the most interesting ones: 1) isometry; 2) geodesic equivalence; 3) orbital equivalence; 4) Liouville equivalence.

Contents Introduction §1. The general Maupertuis principle §2. The Maupertuis principle in the dynamics of a massive rigid body §3. The Maupertuis principle and the explicit form of the metric generated on the sphere by a quadratic Hamiltonian on the Lie algebra of the group of motions of R3 §4. Classical cases of integrability in rigid body dynamics and the corresponding geodesic flows on the sphere §5. Integrable metrics on the torus and on the sphere §6. Conjectures §7. The complexity of integrable geodesic flows of 1-2-metrics on the sphere and on the torus §8. A rougher conjecture: the complexities of non-singularly integrable metrics on the sphere or on the torus coincide with those of the known integrable 1-2-metrics §9. The geodesic flow on an ellipsoid is topologically orbitally equivalent to the Euler integral case in the dynamics of a rigid body Bibliography

In this paper new orbital invariants of integrable Hamiltonian systems with two degrees of freedom are described, considered on non-singular three-dimensional constant-energy surfaces. A classification up to orbit-preserving homeomorphisms is obtained for dynamical systems that describe the rotation of a rigid body around its centre of mass for various values of the parameters.

A classification of the integrable Hamiltonian systems with two degrees of freedom on three-dimensional constant energy surfaces is obtained up to homeomorphisms that preserve the trajectories.Bibliography: 16 titles.

This paper is a natural extension and continuation of the authors' studies of the astronomical dating problem of Ptolemy's famous Almagest. In previous papers, the authors suggested and developed a new geometrical-statistical method for dating ancient star catalogues. This method was then applied to Ptolemy's Almagest. The results obtained do not confirm the traditional dating of the Almagest (2nd century AD or 2nd century BC) but shift it to the epoch AD 600-1300. In this paper, we extend our analysis to other parts of the Almagest and study the dating problem for series of lunar eclipses described in the Almagest and for the covering of stars by planets. The results obtained completely agree with our previous results and give the same time interval, AD 600-1300. Mathematics Subject Classifications (1991). 01A35, 51 F99, 62J99.

Some new objects, bordisms of integrable systems, are found and studied. The classes of rigidly bordant systems form a nontrivial abelian group, which makes it possible to construct new integrable systems on the basis of previously known ones. Among the generators of this bordism group are known physical integrable systems, as, for example, the Lagrange system (from the dynamics of a heavy rigid body) and others. Moreover, a new topological invariant of systems with many degrees of freedom is also constructed. It turns out that two integrable systems are topologically equivalent if and only if their invariants coincide. In particular, it follows from this that the set of topological classes of integrable systems is discrete. The invariant can be effectively calculated for concrete integrable systems arising in physics and mechanics.

A new topological invariant is constructed which classifies integrable Hamiltonian systems with two degrees of freedom (admitting a Bott integral). A criterion for the equivalence of Bott systems is proved: such systems are topologically equivalent if and only if their topological invariants coincide. The topological invariant is effectively calculated for specific integrable Hamiltonian systems in physics and mechanics.

CONTENTS § 1. Introduction § 2. Realizability theorem § 3. Tagged skeletons § 4. Coding integrable Hamiltonian systems References