Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems

Mitsuru Shibayama

doi:10.3934/dcds.2017116

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2017, Volume 37, Issue 5: 2705-2715. Doi: 10.3934/dcds.2017116

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Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems

Mitsuru Shibayama

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku Kyoto 606-8501, Japan

* Corresponding author
* Corresponding author

Received: April 2016

Revised: December 2016

Published: May 2017

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Abstract

We study the existence of periodic solutions for a prescribed-energy problem of Hamiltonian systems whose potential function has a singularity at the origin like $-1/|q|^{α} (q ∈ \mathbb{R}^N)$ . It is known that there exist generalized periodic solutions which may have collisions, and the number of possible collisions has been estimated. In this paper we obtain a new estimation of the number of collisions. Especially we show that the obtained solutions have no collision if $N ≥ 2$ and $α >1$ .

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Mathematics Subject Classification: Primary: 70H12, 70F05; Secondary: 34C25.

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Figure 1. Graph of $f(1, \alpha, \alpha, \alpha)$.

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References

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