Multiple Rotation Type Solutions for Hamiltonian Systems on $T^\ell\times\mathbb{R}^{2n-\ell}$

TL;DR

This paper proves the existence of multiple rotation type solutions for Hamiltonian systems on tori and Euclidean spaces, including at least n+1 solutions for spatially periodic cases and on energy hypersurfaces.

Contribution

It establishes new multiplicity results for Hamiltonian systems on mixed tori and Euclidean spaces, extending known solutions to broader classes.

Findings

At least n+1 rotation solutions for spatially periodic Hamiltonian systems.

Existence of at least one periodic or n+1 rotation solutions on energy hypersurfaces.

Results apply to systems with various torus dimensions, including non-n cases.

Abstract

This paper deals with multiplicity of rotation type solutions for Hamiltonian systems on $T^\ell\times \mathbb{R}^{2n-\ell}$. It is proved that, for every spatially periodic Hamiltonian system, i.e., the case $\ell=n$, there exist at least $n+1$ geometrically distinct rotation type solutions with given energy rotation vector. It is also proved that, for a class of Hamiltonian systems on $T^\ell\times\mathbb{R}^{2n-\ell}$ with $1\leqslant\ell\leqslant 2n-1$ but $\ell\neq n$, there exists at least one periodic solution or $n+1$ rotation type solutions on every contact energy hypersurface.