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

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 $1⩽\ell ⩽2n-1$ but $\ell \ne n$, there exists at least one periodic solution or $n+1$ rotation type solutions on every contact energy hypersurface.

30 pages, 3 figures