Noether's theorem for quasi conservative mechanical systems

by Prof. Đorđe Mušicki (Matemtical Institute SANU Belgrade)

Wednesday 6 Apr 2011, 18:00 → 19:00 Europe/Belgrade

301 F (Mathematical Institute)

If Lagrange's equations for nonconservative systems by introducing a Lagrangian, which is equal to product of some function of time $f(t)$ and primary Lagrangian, can be reduced to Euler-Lagrange's equations, such mechanical systems are named quasi-conservative ones. The condition that some mom-conservative system can be considered as quasi-conservative one is the existence of at least one particular solution, which results from a system of $n$ differential equations with one unknown function -- the cited function $f(t)$. For such systems the energy relations are studied on the basis of corresponding Lagrange's equations, and it is demonstrated that under certain conditions, the some integrals of motion equivalent to Vujanovic's energy like conservation laws are valid.

In this communication the corresponding energy relations are studied from a different, more general variant, on the basis of the corresponding accommodated Noether's theorem. Such Noether's theorem for quasi-conservative systems is formulated, starting from the total variation of action and the corresponding Lagrange's equations and repeating the usual procedure. It differs from the usual Noether's theorem only by presence of the new Lagrangian, extended from the primary one by the function $f(t)$ and by means of which the corresponding condition for the existence of the integrals of motion (the so-called basic Noether's identity) is formulated. It is transformed to a more suitable form, from which under certain conditions the corresponding integrals of motions are obtained, and for their existence, it is necessary that at least one particular solution of one partial differential equation exists. The obtained results are in full accordance with the results obtained on the basis of Lagrange's equations, and so modified Noether's theorem is equivalent to Vujanović-Djukić's formulation of this theorem for nonconservative systems, obtained by transformation of D'Alambert-Lagrange's principle.