Symmetries and conserved quantities for systems of generalized classical mechanics

Abstract

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether’s theorem and the generalized Noether’s inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie’s theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed. PACC: 0320; 1110; 0220 I. INTRODUCTION The conservation laws (or first integrals) of the mechanical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principles. The fact that the existence of a conserved quantity means that an inner dynamical symmetry in the mechanical system plays an irreplaceable role in the physical interpretation of motion. The modern methods in finding conservation laws are mainly two ways [1] with each having a concept of its own: the first one is based upon the invariant properties of Hamiltonian action with respect to the action of the infinitesimal transformation of the finite transformation group, i.e., Noether symmetries and conserved quantities; the second one is based upon the Lie’s theory of the invariance of differential equations under the action of infinitesimal groups. The differential equations of many physical problems appear to be the variational problems. The theory of the generalized classical mechanics, initiated by Ostrogradsky and Jacobi in the years from 1848 to 1858, has been gaining much development during the recent fifty years, and many important results have been made. [2 6] The Lagrange equations with high