# On transversal connecting orbits of Lagrangian systems in non-stationary force field: Newton-Kantorovich approach

## TL;DR

This paper investigates the existence of special connecting orbits in a non-stationary Lagrangian system on a Riemannian manifold, using a Newton-Kantorovich approach to extend geodesics across the entire real line.

## Contribution

It introduces a Newton-Kantorovich method to establish the existence of transversal connecting orbits in non-stationary Lagrangian systems with specific potential and manifold conditions.

## Key findings

- Existence of transversal doubly asymptotic trajectories under certain conditions.
- Conditions on the potential and manifold ensuring orbit existence.
- Extension of geodesics to connect critical points across time.

## Abstract

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to the action of a non-stationary force field with potential $U(q,t) = f(t)V(q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to \pm\infty$ and vanishes at a unique point $t_{0}\in \mathbb{R}$. 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 $t> t_{0}$ and $t<t_{0}$, respectively. Under nondegeneracy conditions on points of $X_{\pm}$ we apply 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 obatained by continuation of geodesics defined in a vicinity of the point $t_{0}$ to the whole real line.