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约束Herglotz方程的代数结构与非保守系统的Poisson积分理论

ALGEBRAIC STRUCTURE AND POISSON THEORY OF HERGLOTZ EQUATIONS FOR NONCONSERVATIVE SYSTEMS

  • 摘要: 无论完整或非完整约束系统, 如果存在非保守力, 则这些系统的方程一般不具有Lie代数结构, 从而经典Poisson积分理论只能部分地应用于这些系统的积分问题. 通过Herglotz原理可导出非保守系统的一类动力学方程, 即Herglotz方程. 研究Herglotz方程的代数结构, 进而建立其Poisson积分理论, 对于深入探寻非保守系统的动力学特性具有重要意义. 文章研究Herglotz方程的代数结构, 进而建立非保守系统的Poisson理论, 包括完整和非完整情形. 首先, 针对完整非保守系统, 建立其Herglotz方程, 引入积分因子将方程化为逆变代数形式, 证明其具有Lie代数结构, 从而Poisson理论可全部地应用于该系统. 其次, 针对非完整非保守系统, 建立其约束Herglotz方程, 利用积分因子将方程化为部分正则的逆变代数形式, 证明约束Herglotz方程具有Lie容许代数结构, 进而建立非完整非保守系统的Poisson理论. 若非完整非保守系统实现自由运动, 则约束Herglotz方程具有Lie代数结构, Poisson理论仍可全部地应用于该系统. 文中以某非线性方程系统, 受均匀和各向同性瑞利耗散力作用的在粗糙水平面上作纯滚动的圆球, 以及受黏性阻尼的Appell-Hamel问题为例, 分析了约束Herglotz方程的代数结构并演示所述Poisson理论的应用.

     

    Abstract: The equations of holonomic or nonholonomic constrained systems generally do not have a Lie algebraic structure if there are nonconservative forces, so the classical Poisson integral theory can only be partially applied to the integral problems of these systems. A class of dynamic equations, i.e., Herglotz’s equations, for nonconservative systems can be derived by Herglotz principle. Studying the algebraic structure of Herglotz equation and establishing its Poisson integral theory is of great significance for exploring the dynamics of nonconservative systems. In this paper, the algebraic structure of Herglotz equations is studied, and then Poisson theory for nonconservative systems is established, including holonomic and nonholonomic cases. Firstly, the Herglotz equations for holonomic nonconservative systems are established, and the equations are transformed into contravariant algebraic form by introducing integral factor. The equations are proved to have Lie algebraic structure, and the Poisson theory can be applied to the system completely. Secondly, for nonholonomic nonconservative systems, the constrained Herglotz equations are established, and the equations are transformed into partially canonical contravariant algebraic form by using integral factor. It is proved that the constrained Herglotz equations have a Lie admissibility algebraic structure, and Poisson theory is established. If the nonholonomic nonconservative system realizes free motion, then the constrained Herglotz equations have a Lie algebraic structure, and Poisson's theory can still be fully applied to the system. In this paper, we analyze the algebraic structure of constrained Herglotz equations and demonstrate the application of the obtained Poisson theory with the three examples, i.e., a nonlinear equation system, a ball rolling on a rough horizontal plane under uniform and isotropic Rayleigh dissipative forces, and the Appell-Hamel problem under viscous damping.

     

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