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

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

2010-03-21

arxiv.org/abs/1003.4035v1

Mathematics

Symplectic Geometry

30 pages, 3 figures

Scientific paper

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.

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