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We consider a natural Hamiltonian system with two degrees of freedom and Hamiltonian $H=\|p\|^2/2+V(q)$. The configuration space $M$ is a closed surface (for noncompact $M$ certain conditions at infinity are required). It is well known that if the potential energy $V$ has $n>2\chi(M)$ Newtonian singularities, then the system is not integrable and has positive topological entropy on energy levels $H=h>\sup V$. We generalize this result to the case when the potential energy has several singular points $a_j$ of type $V(q)\sim -d(q,a_j)^{-\alpha_j}$. Let $A_k=2-2k^{-1}$, $k=2,3,\dots$, and let $n_k$ be the number of singular points with $A_k\le \alpha_j2\chi(M), $$ then the system has a compact chaotic invariant set of noncollision trajectories on any energy level $H=h>\sup V$. This result is purely topological: no analytical properties of the potential, except the presence of singularities, are involved. The proofs are based on the generalized Levi-Civita regularization and elementary topology of coverings. As an example, the plane $n$ center problem is considered.

In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincar\'e., Comment: arXiv admin note: text overlap with arXiv:1606.06708

In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit.

Anti-integrable limit is one of convenient and relatively simple methods for construction of chaotic hyperbolic invariant sets in Lagrangian, Hamiltonian and other dynamical systems. We discuss the most natural context of the method -- discrete Lagrangian systems. Then we present examples and applications.

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar\'e second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu\to 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system.

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincar\'e second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincar\'e only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.

In his study of periodic orbits of the 3 body problem, Hill obtained a formula relating the characteristic polynomial of the monodromy matrix of a periodic orbit and an infinite determinant of the Hessian of the action functional. A mathematically correct definition of the Hill determinant and a proof of Hill's formula were obtained later by Poincar\'e. We give two multidimensional generalizations of Hill's formula: to discrete Lagrangian systems (symplectic twist maps) and continuous Lagrangian systems. We discuss additional aspects which appear in the presence of symmetries or reversibility. We also study the change of the Morse index of a periodic trajectory after the reduction of order in a system with symmetries. Applications are given to the problem of stability of periodic orbits.