变加速动力学系统的广义高斯最小拘束原理

变加速运动在日常生活和工程问题中普遍存在.变加速动力学又称牛顿猝变动力学,因其在混沌理论和非线性动力学中的应用而获得广泛关注.高斯原理是一个具有极值性质的微分变分原理.因此,研究变加速动力学系统的广义高斯原理在理论和应用两方面都有重要意义.文章提出并研究变加速动力学系统的广义高斯原理.首先,引入急动度空间的广义高斯变分概念,将质点的达朗贝尔原理对时间求导数后与广义高斯变分点乘,并利用高斯意义下的理想约束条件,建立了变加速动力学系统的广义高斯原理.在此基础上,通过构造广义拘束函数建立并证明变加速动力学系统的广义高斯最小拘束原理,并给出原理的阿佩尔形式、拉格朗日形式和尼尔森形式.其次,研究原理对变质量力学的推广.从密歇尔斯基方程出发,将它对时间求导并与广义高斯变分点乘,建立了具有理想约束的变质量变加速动力学系统的广义高斯原理.通过构造变质量系统的广义拘束函数,建立并证明变质量力学系统变加速运动的广义高斯最小拘束原理.文中以开普勒-牛顿空间问题为例,利用所得的广义高斯最小拘束原理方法进行计算,验证了方法的有效性.

GENERALIZED GAUSS PRINCIPLE OF LEAST COMPULSION FOR VARIABLE-ACCELERATION DYNAMICAL SYSTEMS

The motion with variable acceleration is common in daily life and engineering problems. Variable acceleration dynamics, also known as Newtonian jerky dynamics, has gained wide attention due to its application in chaos theory and nonlinear dynamics. Gauss principle is a differential variational principle with extreme value characteristics. Therefore, it is of great significance to study the generalized Gauss principle of dynamical systems with variable acceleration in both theory and application. In this paper, the generalized Gauss principle for dynamical systems with variable accelerated motion is presented and studied. Firstly, we introduce the concept of the generalized Gaussian variation in the jerky space. We take the derivative of d’Alembert principle of a particle with respect to time, and then calculate its dot product with the generalized Gaussian variation. By using the condition of ideal constraints in the sense of Gauss, we establish the generalized Gauss principle for dynamical systems with variable acceleration. On this basis, the generalized Gauss principle of least compulsion is established and proved by constructing the generalized compulsion function. The Appell form, Lagrange form and Nielsen form of the principle are given. Secondly, the extension of the principle to variable mass mechanics is explored. Starting from Meshchersky equation and taking its derivative with respect to time, and then calculating its dot product with the generalized Gaussian variation, we establish the generalized Gauss principle for variable-mass variable-acceleration dynamical systems with ideal constraints. The generalized compulsion function in the case of variable mass is constructed and the generalized Gauss principle of least compulsion for variable-mass variable-acceleration mechanical systems is established and proved. We take the Kepler-Newton problem as an example, and use the approach of the generalized Gauss least compulsion principle we presented to calculate, and verify the effectiveness of the method.