# Rotating Quasi-periodic Solutions of Second Order Hamiltonian Systems with Sub-quadratic Potential

## TL;DR

This paper establishes the existence of multiple rotating quasi-periodic solutions in second order Hamiltonian systems with sub-quadratic potentials, introducing a new index to estimate their number.

## Contribution

It introduces the $ ext{Q}(s)$ index, a novel tool for analyzing rotating quasi-periodic solutions in Hamiltonian systems with sub-quadratic potentials.

## Key findings

- Proves existence of multiple rotating quasi-periodic solutions.
- Develops the $ ext{Q}(s)$ index for solution estimation.
- Provides bounds on the number of such solutions.

## Abstract

This paper concerns the existence of multiple rotating quasi-periodic solutions for second order Hamiltonian systems with sub-quadratic potential. Such solutions have the form $x(t+T)=Qx(t)$ for some orthogonal matrix $Q$. To deal with such quasi-periodic solutions, we introduce the $\mathcal{Q}(s)$ index which is a development of the well known $S^1$ index. Applying the $\mathcal{Q}(s)$ index, we give an estimate of the number for rotating quasi-periodic orbits with a fixed period.