A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(\mathbb{T}^2\)
type: Item
instance of: scholarly article
title: A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(\mathbb{T}^2\)
zbMATH Open document ID: 0952.70015
DOI: 10.1007/PL00020961
author: Amadeu Delshams
author: Rafael de la Llave
author: Tere M. Seara
published in: Communications in Mathematical Physics
publication date: 2000-05-11T00:00:00Z
review text: The main result can be formulated as follows: Let \(g\) be a \(C^r\) generic metric on \(\mathbb{T}^2\), \(U: \mathbb{T}^2\times \mathbb{T}\to \mathbb{R}\) be a \(C^r\) generic function, and \(r\) be sufficiently large. Then Euler-Lagrange equations for the Lagrangian \(L(q,\dot q,t)={1\over 2}g(\dot q,\dot q)- U(q,t)\) have solution \(q(t)\) with energy \(E(t)= {1\over 2} g(\dot q(t),\dot q(t))+ U(q(t), t)\) tending to infinity as \(t\to\infty\). In fact, the authors prove a much more general theorem which ensures the existence of orbits whose energy makes arbitrary excursions for a specified genericity, \(r\geq 15\) is certainly sufficient. The proof is based on subtle geometric perturbations and uses a wide scale of techniques: Riemannian geometry to determine the periodic orbits, the hyperbolicity theory of invariant manifolds, Hamiltonian perturbations, a qualitative version of KAM theory, Melnikov integrals, Poincaré function, and so on. The article also contains some valuble remarks on the Arnold's diffusion and on the Mather variational methods, and numerous comments on the current literature.
Mathematics Subject Classification ID: 70H09
Mathematics Subject Classification ID: 37N05
zbMATH DE Number: 1422925
zbMATH Keywords: orbits with unbounded energy
zbMATH Keywords: geodesic flow
zbMATH Keywords: generic metric
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: Lagrangian
zbMATH Keywords: geometric perturbations
zbMATH Keywords: Riemannian geometry
zbMATH Keywords: periodic orbits
zbMATH Keywords: hyperbolicity theory of invariant manifolds
zbMATH Keywords: Hamiltonian perturbations
zbMATH Keywords: KAM theory
zbMATH Keywords: Melnikov integrals
zbMATH Keywords: Poincaré function
zbMATH Keywords: Arnold's diffusion
zbMATH Keywords: Mather variational methods
reviewed by: Jan Chrastina
MaRDI profile type: Publication
full work available at URL: pl00020961
OpenAlex ID: W2061251427
description: scientific article; zbMATH DE number 1422925