# American Institute of Mathematical Sciences

May  2003, 9(3): 603-616. doi: 10.3934/dcds.2003.9.603

## Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces

 1 Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092, Tunis, Tunisia

Received  December 2001 Revised  November 2002 Published  February 2003

We study the existence of periodic solutions of the first order Hamiltonian system

$\dot q = H_p (p,q),\quad \dot p=-H_q(p,q),$

such that

$H(p,q)= h,$

when the prescribed energy surface $S_h=${$(p,q)\in \mathbf R^N \times \mathbf R^N;H(p,q)=h$} is non-compact.
In our previous work, we have considered the class of singular Hamiltonians like

$H(p,q)$~$(|p|^\beta /\beta )-(1 /|q|^\alpha) \quad$ with $1 \leq\alpha<\beta$ and $\beta\geq 2.$

It has proven the existence of generalized (collision) solutions as a limit of approximate solutions corresponding to critical points of certain functionals. In this paper, we relate the Morse index of critical points with the number of collisions of the generalized solution via blow up arguments. In particular, we obtain the existence of a classical (non-collision) solution for $\alpha \in ]\beta /2,\beta$[ when $N \geq 4$ and for $\alpha \in ]2\beta/ 3 ,\beta$[ when $N=3$. As a consequence, we get for smooth Hamiltonians like

$H(p,q)$~$|q|^\alpha (|p|^\beta +1) \quad$ with $1< \alpha < \beta$ and $\beta \geq 2,$

the same existence results since the two classes of Hamiltonians have the same energy surfaces.

Citation: Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603
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