可积旋力哈密顿量：双哈密顿结构、可分性与周期轨道
Integrable curl-force Hamiltonians: bi-Hamiltonian structure, separability, and periodic orbits
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中文总结 AI 辅助
本文研究具有不定动能的哈密顿旋力系统，通过分析Berry模型的Painlevé可积性，引入四参数旋力族并构造双哈密顿结构等，揭示闭合轨迹不代表可积性的结论。
中文摘要 AI 辅助
我们研究具有不定动能的哈密顿旋力系统。首先重新考察Berry的多项式哈密顿旋力模型，其数值观测到的闭合轨迹推动了可积性猜想。Painlevé分析得到非主共振谱，因此对应的Laurent级数无法容纳通解所需的任意常数数量，该模型未通过标准Painlevé检验。随后我们引入四参数旋力族，确定该系统可积的参数轨迹，构造出第二个哈密顿量、相容泊松张量、分离的复特征变量及Lax表示。更一般地，分离形式可产生任意次数的多项式可积旋力哈密顿量。我们还证明，同一构造允许更高阶时间导数的势化，其自由极限是退化Pais-Uhlenbeck振子。最后分析零旋不变约化与椭圆周期解，展示了可积区域外的孤立周期轨道，说明仅闭合轨迹不意味着Liouville或Painlevé可积性。
英文摘要
We investigate Hamiltonian curl-force systems with indefinite kinetic energy. We first reconsider Berry's polynomial Hamiltonian curl-force model, whose numerically observed closed trajectories motivated an integrability conjecture. A Painlevé analysis yields a non-principal resonance spectrum, so that the corresponding Laurent series cannot accommodate the required number of arbitrary constants of the general solution. The model therefore fails the standard Painlevé test. We then introduce a four-parameter curl-force family and identify the parameter locus on which this system is integrable. We construct a second Hamiltonian, compatible Poisson tensors, separated complex characteristic variables, and a Lax representation. More generally, the separated form yields polynomial integrable curl-force Hamiltonians of arbitrary degree. We also show that the same construction admits a higher time-derivative potentialisation whose free limit is the degenerate Pais-Uhlenbeck oscillator. Finally, we analyse zero-curl invariant reductions and elliptic periodic solutions, and exhibit an isolated periodic orbit outside the integrable regime. This illustrates that closed trajectories alone do not imply Liouville or Painlevé integrability.
