Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems

This paper concerns the existence of multiple rotating periodic solutions for $2n$ dimensional convex Hamiltonian systems. For the symplectic orthogonal matrix $Q$, the rotating periodic solution has the form of $z(t+T)=Qz(t)$, which might be periodic, anti-periodic, subharmonic or quasi-periodic according to the structure of $Q$. It is proved that there exist at least $n$ geometrically distinct rotating periodic solutions on a given convex energy surface under a pinched condition, so our result corresponds to the well known Ekeland and Lasry's theorem on periodic solutions. It seems that this is the first attempt to solve the symmetric quasi-periodic problem on the global energy surface. In order to prove the result, we introduce a new index on rotating periodic orbits.Comment: arXiv admin note: text overlap with arXiv:1812.0583

Global surfaces of section for Reeb flows in dimension three and beyond

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written for Proceedings of the ICM, Rio 2018; in v3 we made minor additional corrections, updated references, added a reference to work of Lu on the Conley Conjectur

Global surfaces of section in the planar restricted 3-body problem

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The approach relies on recent global methods in symplectic geometry and contrasts sharply with the perturbative methods used until now.Comment: 11 pages, 1 figur

Linear stability in billiards with potential

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure

Stability Properties of the Riemann Ellipsoids

We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Riemann ellipsoids, except a finite number of codimension one subfamilies, are Nekhoroshev--stable. We base our analysis on a Hamiltonian formulation of the problem on a covering space, using recent results from Hamiltonian perturbation theory.Comment: 29 pages, 6 figure

Aspects of Discrete Breathers and New Directions

We describe results concerning the existence proofs of Discrete Breathers (DBs) in the two classes of dynamical systems with optical linear phonons and with acoustic linear phonons. A standard approach is by continuation of DBs from an anticontinuous limit. A new approach, which is purely variational, is presented. We also review some numerical results on intraband DBs in random nonlinear systems. Some non-conventional physical applications of DBs are suggested. One of them is understanding slow relaxation properties of glassy materials. Another one concerns energy focusing and transport in biomolecules by targeted energy transfer of DBs. A similar theory could be used for describing targeted charge transfer of nonlinear electrons (polarons) and, more generally, for targeted transfer of several excitations (e.g. Davydov soliton).Comment: to appear in the Proceedings of NATO Advanced Research Workshop "Nonlinearity and Disorder: Theory and Applications", Tashkent,Uzbekistan,October 1-6, 200

Ehrenfest regularization of Hamiltonian systems

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stable in physical reality (as demonstrated by the unexpected change of rotation of the Explorer 1 probe). We propose a general method of Ehrenfest regularization of Hamiltonian equations by which the reversible Hamiltonian equations are equipped with irreversible terms constructed from the Hamiltonian dynamics itself. The method is demonstrated on harmonic oscillator, rigid body motion (solving the problem of stable minor axis rotation), ideal fluid mechanics and kinetic theory. In particular, the regularization can be seen as a birth of irreversibility and dissipation. In addition, we discuss and propose discretizations of the Ehrenfest regularized evolution equations such that key model characteristics (behavior of energy and entropy) are valid in the numerical scheme as well