Physical consequences of action conservation laws and their applications

Abstract

A class of conservation laws containing Hamilton's action integral is introduced for Lagrangian dynamical systems with a single degree of freedom and for the case when the Lagrangian function depends on the second time derivative of the coordinate. The action conservation laws are derived from the invariant properties of the Lagrange-D'Alembert differential variational principle with respect to infinitesimal transformations of the generalized coordinate and time by supposing that the generators of infinitesimal transformations depend on time, a generalized coordinate, and its first and second derivatives with respect to time. These action integral conservation laws are applied to the stability of columns, heat transfer, Thomas-Fermi problems, and other physical phenomena. A direct method for the approximate solution of these problems is combined with the Ritz variational method in order to obtain results of high accuracy.