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.