基于非标准Lagrange函数的动力学系统的广义能量积分与Whittaker降阶法

Generalized Energy Integral and Whittaker Method of Reduction for Dynamics Systems with Non-standard Lagrangians

研究基于非标准Lagrange函数的动力学系统的广义能量积分和Whittaker降阶方法。首先,基于指数Lagrange函数和Lagrange函数幂函数两类非标准Lagrange函数,定义了相应的Hamilton作用量,建立了该系统的Hamilton原理,给出了系统的Lagrange方程。其次,利用系统的Lagrange方程,建立了基于非标准Lagrange函数的广义能量积分存在的条件及形式。然后,将著名的Whittaker降阶法加以推广,得到了基于非标准Lagrange函数的动力学系统的Whittaker方程。最后,以算例验证了本文结果。

The generalized energy integral and Whittaker method of reduction for the dynamics system based on non-standard Lagrangians are studied. Firstly, in view of two kinds of non-standard Lagrangians, i. e. , exponential Lagrangians and power law Lagrangians, the Hamilton action with non-standard Lagrangians is defined, and the Hamilton principles and the Lagrange equations of the system are ob- tained. Secondly, the condition under which the generalized energy integral with non-standard Lagrang- ians exists and the form of generalized energy integral are established by using the Lagrange equations of the system. Thirdly, the famous Whittaker method of reduction is extended, and the Whittaker equa- tions for the dynamics system with non-standard Lagrangians are obtained. Finally, an example is given to illustrate the application of the results.