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 引用本文: 宋传静, 侯爽. 时间尺度上约束力学系统的Noether型绝热不变量. 力学学报, 2024, 56(8): 2397-2407.
Song Chuanjing, Hou Shuang. Noether-type adiabatic invariants for constrained mechanics systems on time scales. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(8): 2397-2407.
 Citation: Song Chuanjing, Hou Shuang. Noether-type adiabatic invariants for constrained mechanics systems on time scales. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(8): 2397-2407.

## NOETHER-TYPE ADIABATIC INVARIANTS FOR CONSTRAINED MECHANICS SYSTEMS ON TIME SCALES

• 摘要: 时间尺度微积分是近年来的研究热点之一, 其不仅统一连续分析与离散分析, 还能协助完成更复杂动力学系统的建模. Noether对称性方法是一种近代的积分方法, 揭示了力学系统守恒量与其内在的动力学对称性之间的潜在关系, Noether对称性的摄动与绝热不变量和系统的可积性之间也有着密切的联系. 时间尺度上约束力学系统的Noether对称性问题虽然已有学者研究, 但是由于时间尺度微积分理论的不成熟, 研究成果的深度及正确性均有待探究. 文章的重点是探讨时间尺度上约束力学系统Noether对称性的摄动与绝热不变量, 这其中包含了Lagrange系统、Hamilton系统以及Birkhoff系统. 首先, 探讨了3个受小扰动作用的系统, 其Noether对称性的变化, 并给出了相应的绝热不变量; 然后提供了在无扰动条件下, 3个系统的精确不变量. Lagrange系统得到的精确不变量与原有结果吻合, Hamilton系统和Birkhoff系统得到的精确不变量是新的. 其次, 时间尺度上的导数分为delta导数和nabla导数, 由这两种导数所组成的系统, 互为对偶系统. 基于文章delta导数下所得到的结果, 采用对偶原理的方法, 给出了3个系统对偶空间的绝热不变量和精确不变量. 最后, 文末分别讨论了时间尺度上Kepler问题和Hojman-Urrutia问题的Noether型绝热不变量, 从而借助例题对3个系统中所得到的结果和所采用的方法进行说明.

Abstract: Time scale calculus is one of the research hotspots in recent years. It not only unifies continuous analysis and discrete analysis, but also assists in modeling more complex dynamics systems. The Noether symmetry method is a modern integration method that reveals the potential relationship between the conserved quantity of a mechanics system and its inherent dynamics symmetry. Perturbation to Noether symmetry, as well as adiabatic invariants, are also closely related to the integrability of the system. Although the problems of symmetry for constrained mechanics systems on time scales have been studied by scholars, the depth and accuracy of research results need to be explored due to the immaturity of time scale calculus theory. The focus of this article is to explore the perturbation to Noether symmetry, and the adiabatic invariants for the constrained mechanics systems on time scales, including Lagrangian system, Hamiltonian system and Birkhoffian system. Firstly, for the three perturbed systems, we discuss the changes of the Noether symmetry, and present the corresponding adiabatic invariants. Then, we provide the exact invariants of the three systems under disturbance free conditions. The exact invariant obtained of the Lagrangian system is consistent with the original result, while the exact invariants obtained of the Hamiltonian and Birkhoffian systems are new. Secondly, there are two derivatives on time scales, namely, the delta derivative and the nabla one, and the systems composed of the two derivatives are dual. Based on the results obtained under the delta derivative in this article, the adiabatic invariants and exact invariants of the three dual spaces are given using the method of dual principle. Thirdly, at the end of the article, the Noether type adiabatic invariants of the Kepler problem and the Hojman-Urrutia problem on time scales were discussed respectively, to illustrate the results and methods presented in the three systems of this article by examples.

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