A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(\mathbb{T}^2\)
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.
hyperbolicity theory of invariant manifolds
scientific article; zbMATH DE number 1422925