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A prescribed energy problem for a singular Hamiltonian system with a weak force. (English) Zbl 0771.70014
We consider the existence of periodic solutions of a Hamiltonian system $$\ddot q+\nabla V(q)=0$$ such that $$1/2\mid \dot q(t)\mid^ 2+V(q(t))=H$$ for all $$t$$, where $$q\in \mathbb{R}^ N(N\geq 3)$$, $$H<0$$ is a given number, $$V(q)$$ is a potential with a singularity, and $$\nabla V(q)$$ denotes its gradient. In a particular case we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under some assumption we estimate the number of collisions of generalized solutions and get the existence of a classical (non-collision) solution.

##### MSC:
 70H05 Hamilton’s equations 37G99 Local and nonlocal bifurcation theory for dynamical systems
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