标题： 非共振霍普夫环近似哈密顿平衡点 显示英文标题

标题： Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point

摘要： 本文讨论的是二维自由度哈密顿系统在平衡点附近的周期轨道的存在性。 假设该平衡点是哈密顿量的非退化极小值。 足够接近平衡点的能面的每个类似球面的组成部分至少包含两个形成Hopf链环的周期轨道（A. Weinstein [19]）。 Hofer、Wysocki和Zehnder [9] 的一个定理表明，这样的能面的组成部分上要么恰好有两个，要么有无限多个周期轨道。 这一多重性结果来自于存在一个类似圆盘的全局截面。 如果满足Hopf链环轨道旋转数的某种非共振条件 [8]，则会得到无限多个周期轨道。 本文旨在提出哈密顿函数在平衡点处的Birkhoff-Gustavson正规形式的显式条件，通过检查如[8]中的非共振条件来确保能面上存在无限多个周期轨道，而不使用任何全局截面。 主要结果集中在强共振平衡点上，并适用于空间等腰三体问题、Hill的月球问题和Henon-Heiles系统。 显示英文摘要

摘要： 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 H\'enon-Heiles System.