On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach

Abstract

We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t₀ ∈ ℝ. Let X₊, X₋ denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t > t₀ and t < t₀, respectively. Under nondegeneracy conditions on points of X_± we apply the Newton – Kantorovich type method to study the existence of transversal doubly asymptotic trajectories connecting X₋ and X₊. Conditions on the Riemannian manifold and the potential which guarantee the existence of such orbits are presented. Such connecting trajectories are obtained by continuation of geodesies defined in a vicinity of the point t₀ to the whole real line.