从特解进行哈密顿约化
Hamiltonian reduction from particular integrals
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中文总结 AI 辅助
研究从特解出发的哈密顿约化，开发由特解生成的几何约化机制，通过一族函数定义不变零级子流形，在特解、预辛约化和低维哈密顿动力学间建桥梁，还引出特解可积性概念，并用例子说明框架。
中文摘要 AI 辅助
我们开发了一种由特解生成的几何约化机制。一族函数，其时间导数在同一族上线性闭合，定义了一个不变零级子流形。在哈密顿情形下，如果这族函数相互对合，受限动力学是预辛的，其特征商承载约化的哈密顿流。这在特解、预辛约化和低维哈密顿动力学之间建立了直接桥梁，并引出了特解可积性的刘维尔型概念。我们通过力学例子和提升构造来说明该框架，包括艾森哈特提升的变体。
英文摘要
We develop a geometric reduction mechanism generated by systems of particular integrals, namely, families of functions whose time derivatives close linearly on the family. Their common zero set is dynamically invariant. In the Hamiltonian case, under a weak involution condition, the restricted dynamics is presymplectic, and its characteristic quotient carries a reduced Hamiltonian flow. This yields a direct bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics, and leads to a Liouville-type notion of particular integrability. We illustrate the framework through mechanical examples and lift constructions, including variants of the Eisenhart lift.
