# Periodic orbits in the restricted three-body problem and Arnold's $J^+$-invariant

## TL;DR

This paper applies Arnold's theory of plane curves to Stark-Zeeman systems, introducing invariants of periodic orbits based on the $J^+$-invariant and analyzing their behavior in these Hamiltonian systems.

## Contribution

It extends Arnold's $J^+$-invariant to Stark-Zeeman systems, providing new invariants for periodic orbits and studying their properties.

## Key findings

- Introduces invariants of periodic orbits in Stark-Zeeman systems
- Analyzes the behavior of these invariants under system dynamics
- Connects Arnold's theory to celestial mechanics applications

## Abstract

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.