Sergey Bolotin

We consider Hamiltonian systems possessing families of nonresonant invariant tori whose frequencies are all collinear. Then under certain conditions the frequencies depend on energy only. This is a generalization of the well-known Gordon’s theorem about periodic solutions of Hamiltonian systems. While the proof of Gordon’s theorem uses Hamilton’s principle, our result is based on Percival’s variational principle. This work was motivated by the problem of isochronicity in Hamiltonian systems.

We consider a Hamiltonian system depending on a parameter which slowly changes with rate $\varepsilon \ll 1$. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order $\varepsilon$. We prove a partial analog of Neishtadt's result for a system with $n$ degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order $\varepsilon$ at time intervals of order $|\ln\varepsilon|$, so the energy may grow with rate $\varepsilon/|\ln\varepsilon|$. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order $\varepsilon$.

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 Poincaré.

Let $M$ be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose $M$ consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the $\lambda$-lemma) describing the behavior of trajectories near $M$. Using this result, trajectories shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s 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 $M$ of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem.

We consider a mechanical system inside a rolling ball and show that if the constraints have spherical symmetry, the equations of motion have Lagrangian form. Without symmetry, this is not true.

We study the problem of optimal control of a Chaplygin ball on a plane by means of 3 internal rotors. Using Pontryagin maximum principle, the equations of extremals are reduced to Hamiltonian equations in group variables. For a spherically symmetric ball, the solutions can be expressed in by elliptic functions.

We show that under certain natural conditions the definition of a hyperbolic torus conventional for the general theory of dynamical systems is quite suitable for needs of the KAM-theory.

A time-periodic Hamiltonian system on a cotangent bundle of a compact manifold with Hamiltonian strictly convex and superlinear in the momentum is studied. A hyperbolic Diophantine nondegenerate invariant torus $N$ is said to be minimal if it is a Peierls set in the sense of the Aubry–Mather theory. We prove that $N$ has an infinite number of homoclinic orbits. For any family of homoclinic orbits the first and the last intersection point with the boundary of a tubular neighborhood $U$ of $N$ define sets in $U$. If there exists a compact family of minimal homoclinics defining contractible sets in $U$, we obtain an infinite number of multibump homoclinic, periodic and chaotic orbits. The proof is based on a combination of variational methods of Mather and a generalization of Shilnikov's lemma.

The results of Morse and Hedlund about minimal heteroclinic geodesics on surfaces are generalized to a class of Finsler manifolds possessing a symmetry. The existence of minimal heteroclinic geodesics is established. Under an assumption that the set of such geodesics has certain compactness properties, multibump chaotic geodesics are constructed.