Noether's theorem for nonconservative systems in quasicoordinates

Djordje Mušicki

In this paper the generalized Noether's theorem is given in quasicoordinates for the systems of particles, the motion of which can be presented in quasicoordinats and quasivelo\-ci\-ties. After a systematic review of the calculus with quasicoordinates and the corresponding Boltzmann--Hamel's equations of motion, the total variation of action is given in quasicoordinates. Then, the corresponding generalized Noether's theorem is formulated, valid for nonconservative systems as well, which is obtained from the total variation of action and corresponding Boltzmann--Hamel's equations. So formulated Noether's theoerm in quasicoordinates is valid for all conservative and nonconservative systems without any limitation. It is applied to obtain the corresponding energy integrals in quasicoordinates for conservative and nonconservative systems, in the latter case these are energy integrals in broader sense. The obtained results are illustrated by a characteristic example, where the corresponding energy integral is found. This generalized Neother's theorem is equivalent, but not in the form and with some limitation, to the corresponding Noether's theorem formulated by Dj. Djukić [13], which is obtained from the invariance of total variation only of element of action $\Delta(L\,dt)$. However, for nonconservative systems the Lagrangian $L$, appearing in this relations, represents not the usual, but an equivalent Lagrangian, which completely determines the considered system, including the influence of nonpotential forces. Therefore, the cited Noether's theorem is valid only for these nonconservative systems for which it is possible to find such equivalent Lagrangian, (what for the natural systems is mostly possible).