Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917946370031616 |
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| author | Grotta-Ragazzo, C. Liu, Lei Salomão, Pedro A. S. |
| author_facet | Grotta-Ragazzo, C. Liu, Lei Salomão, Pedro A. S. |
| contents | This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (A. Weinstein [19]). A theorem by Hofer, Wysocki, and Zehnder [9] implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied [8], then infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in [8] and not making use of any global surface of section. The main results focus on strongly resonant equilibrium points and apply to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Hénon-Heiles System. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point Grotta-Ragazzo, C. Liu, Lei Salomão, Pedro A. S. Dynamical Systems Mathematical Physics Symplectic Geometry This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (A. Weinstein [19]). A theorem by Hofer, Wysocki, and Zehnder [9] implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied [8], then infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in [8] and not making use of any global surface of section. The main results focus on strongly resonant equilibrium points and apply to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Hénon-Heiles System. |
| title | Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point |
| topic | Dynamical Systems Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2503.03226 |