Sergey V. Bolotin

Unbounded growth of energy in nonautonomous Hamiltonian systems

The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincare-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian. AMS classification scheme number: 58F05

Invariant Sets of Hamiltonian Systems and Variational Methods

We study the problem on the existence of homoclinic trajectories to Mather minimizing invariant sets (multidimensional generalization of Aubry-Mather sets) of positive definite time-periodic Hamiltonian systems [19]. These sets are supports of invariant probability measures in the Lagrangian L. For natural systems with L = ǁυǁ2/2 — V(x), the minimizing set is Γ = {V = h}, h = max V, and for time-periodic systems with reversible L the minimizing sets consist of brake orbits of minimal action. For natural Hamiltonian systems, the existence of homoclinics to Γ was proved in [3] using the Maupertuis-Jacobi functional ( smallint sqrt {h - V(x)} parallel dxparallel ,) and for reversible time-periodic systems in [4] using Hamilton’s functional (see also [5], [16]). For nonreversible systems (for example, natural systems with gyroscopic forces), in general there are no Mather sets of simple structure. We extend the above existence results to arbitrary minimizing sets replacing homoclinic trajectories by semihomoclinic ones in Birkhoff’s sense [2]. A similar problem was studied in [20].It has been observed by physicists for a long time that symplectic structures arise naturally from boundary value problems. For example, the Robbin quotient n n

VARIATIONAL CRITERIA FOR NONINTEGRABILITY AND CHAOS IN HAMILTONIAN SYSTEMS

V = {text{dom}}D*/{text{dom}}D,

Asymptotic solutions of Lagrangian systems with gyroscopic forces

n nassociated to a symmetric (but not self-adjoint) operator D: dom D → H on a Hilbert space H carries a symplectic structure n n

Global regularization for the

omega (upsilon ,omega ) = leftlangle {D*upsilon ,left. omega rightrangle - leftlangle {upsilon ,D*omega } rightrangle } right.

-center problem on a manifold

n n.