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 引用本文: 张毅. 非保守广义Chaplygin系统的Herglotz型Noether定理. 力学学报, 待出版.
Zhang Yi. Noether's theorem of herglotz form for generalized chaplygin systems with nonconservative forces. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Zhang Yi. Noether's theorem of herglotz form for generalized chaplygin systems with nonconservative forces. Chinese Journal of Theoretical and Applied Mechanics, in press.

## NOETHER'S THEOREM OF HERGLOTZ FORM FOR GENERALIZED CHAPLYGIN SYSTEMS WITH NONCONSERVATIVE FORCES

• 摘要: Herglotz原理是非保守系统的变分原理, 它是经典Hamilton原理的一个推广. 在非保守情形, 现有推广形式的Hamilton原理尽管也可导出运动方程, 但实际上不再是变分原理. Noether定理是分析力学的一个基本原理, 它阐明了对称性和守恒律的相关性, 利用Noether定理可找到比牛顿力学和分析力学更多的守恒律. Chaplygin系统是一类特殊而又应用广泛的非完整系统, 其方程可独立于约束方程进行研究. 非保守广义Chaplygin系统则是Chaplygin系统对非保守力学的推广, 因此研究建立该系统的Herglotz型Noether定理具有重要意义. 文章针对非完整约束系统, 提出基于Herglotz原理建立非保守广义Chaplygin系统的动力学方程, 利用变分关系以及虚位移的Chetaev条件, 从关于Hamilton-Herglotz作用量的微分方程出发推导建立其全变分方程并求解, 由此导出非保守广义Chaplygin系统的作用量的全变分. 给出Herglotz型Noether对称变换及准对称变换的定义, 在此基础上利用全变分公式导出其判据 (Noether等式), 建立并证明非保守广义Chaplygin系统Herglotz形式的Noether定理, 得到新型守恒量. 最后, 针对两个具体的非保守非完整系统, 建立了它们的Herglotz型广义Chaplygin方程、判据方程和Chetaev条件, 解出生成元, 由文中所得定理找到了系统的守恒量, 验证了结果的有效性.

Abstract: Herglotz principle is a variational principle for nonconservative systems, which is a generalization of Hamilton's principle. In the non-conservative case, Hamilton's principle, in its current generalized form, is no longer a variational principle, although it can also be used to derive equations of motion. Noether theorem is a basic principle of analytical mechanics, which reveals the intrinsic relationship between symmetry and conservation laws. More conservation laws can be found by using Noether theorem than Newtonian mechanics and analytical mechanics. Chaplygin system is a special and widely used nonholonomic system whose equations can be studied independently of the constraint equations. The nonconservative generalized Chaplygin system is a generalization of Chaplygin system to nonconservative mechanics, so it is of great meaning to explore and establish Herglotz type Noether theorem for this system. To aim at the nonholonomic mechanical systems, we proposed and established the dynamical equations of nonconservative generalized Chaplygin systems based on the Herglotz principle. By using variational relation and Chetaev condition of virtual displacement, the total variational equation of Hamilton-Herglotz action is derived from the derivation and solved. From this, the total variation of the action of nonconservative generalized Chaplygin system is derived. The definitions of Herglotz form Noether symmetric transformation and quasi-symmetric transformation are given, and their criterions (Noether identities) are derived by using total variational formula. Noether’s theorems of Herglotz form for generalized Chaplygin systems with nonconservative forces are proved, and a new type of conserved quantity is obtained. Finally, for two specific nonconservative nonholonomic systems, the generalized Chaplygin equations of Herglotz form, the criterion equations and Chetaev conditions are set up, and the generators are solved, and conserved quantities are found by the theorems obtained, and the validity of the results is verified.

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