Conservation laws and Noether's theorem in a parametric formulation of mechanics

Đ. Mušicki

Theoretical and applied mechanics, Tome 19 (1993) no. 1, p. 87 .

Résumé

In this article we have studied the conservation laws in a parametric formulation of rheonome system mechanics, which is based on a family of varied trajectories, drawn from their initial position and on the transition to a new parameter, which depends on the chosen trajectory. First are analyzed the various forms of the energy law, obtained using the differential equations of motion, in usual and this formulation of mechanics. Furthermore, here we have formulated a general criterion for the integrals of motion, expressed by means of extended Poisson brackets. In the second part of this article we obtained the generalized Noether theorem in this parametric formulation of mechanics, from both the corresponding principle of d'Alembert and Lagrange, as well as from the total variation of action. Applying this to these rheonome systems, the energy law is found in the form $\varepsilon=T+U+P=\mathrm{const.}$, where $P$ is the rheonome potential, as well as using the extended system of Lagrange equations. These results are in agreement with those obtained by V. Vujičić in his modification of the mechanics of rheonome systems.