Periodic orbits of a one dimensional non autonomous Hamiltonian system

TL;DR

This paper investigates the existence and properties of periodic orbits in a one-dimensional non-autonomous Hamiltonian system with periodic potential, establishing the existence of infinitely many solutions and providing bounds using Morse theory.

Contribution

It demonstrates the existence of infinitely many periodic solutions with a given winding number and provides lower bounds on their number using Morse theory.

Findings

Existence of infinitely many periodic orbits for given winding numbers.

Lower bounds on the number of periodic orbits with specific periods and winding numbers.

Application of Morse theory to estimate the number of solutions.

Abstract

In this paper we study the properties of the periodic orbits of \"x + V'_x(t, x) = 0 with x \in S1 and V(t, x) a T0 periodic potential. Called {\rho} \in (1/T0)Q the frequency of windings of an orbit in S1 we show that exists an infinite number of periodic solutions with a given {\rho}. We give a lower bound on the number of periodic orbits with a given period and {\rho} by means of the Morse theory.