On the symmetry approach to polynomial conservation laws of one-dimensional Lagrangian systems

In this paper we analyze polynomial conservation laws of one-dimensional no n-autonomous Lagrangian dynamical systems x = - partial derivative Pi (t,x) /partial derivativex. The analysis is based upon application of Noether's t heorem which relates the existence of conservation laws to the symmetries o f Hamilton's action integral. It is shown that the existence of first integ rals depends on the solution of the system of first-order partial different ial equations - generalized Killing's equations. General solution of the pr oblem is formally determined. It is demonstrated that the final form of dyn amical system and corresponding conservation law depends on the Solution of the so-called potential equation. However. the structure of symmetry trans formations, which generate particular class of conservation laws, could be prescribed independent of the solution of potential equation. This fact is used to underline phenomenological aspect of symmetry approach. Its pragmat ic value is confirmed through several concrete examples. (C) 2001 Elsevier Science Ltd. All rights reserved.