The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere
The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere
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DOI：
10.1515/ans-2016-6003
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发表时间：
2016-08
影响因子：
1.8
通讯作者：
Alberto Abbondandolo;L. Asselle;G. Benedetti;M. Mazzucchelli;I. Taimanov
中科院分区：
文献类型：
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作者：
Alberto Abbondandolo;L. Asselle;G. Benedetti;M. Mazzucchelli;I. Taimanov
摘要考虑2-球面余切丛上的磁Tonelli哈密顿系统，其中磁形式不一定是精确的。众所周知，在很低和很高的能量水平上，这些系统可能只有100多个周期轨道。我们的主要结果表明，在精确表征的中间区（e_0，e_1）{（e_{0}，e_{1}）}中，几乎所有的能级都具有无穷多个周期轨道。这样的能量范围是非空的，例如，在物理相关的情况下，托内利拉格朗日是动能，磁形式是振荡的（在这种情况下，e0=0${e_{0}=0}$是系统的最小能量）。
Abstract We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range (e0,e1)${(e_{0},e_{1})}$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e0=0${e_{0}=0}$ is the minimal energy of the system).