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This paper deals with multiplicity of rotation type solutions for Hamiltoniansystems on $T^\ell\times \mathbb{R}^{2n-\ell}$. It is proved that, for everyspatially periodic Hamiltonian system, i.e., the case $\ell=n$, there exist atleast $n+1$ geometrically distinct rotation type solutions with given energyrotation 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 $n+1$ rotationtype solutions on every contact energy hypersurface.

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