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

Abstract

In this paper we analyze polynomial conservation laws of one-dimensional non-autonomous Lagrangian dynamical systems x ̈ =−∂ Π(t,x)/∂x . The analysis is based upon application of Noether's theorem which relates the existence of conservation laws to the symmetries of Hamilton's action integral. It is shown that the existence of first integrals depends on the solution of the system of first-order partial differential equations — generalized Killing's equations. General solution of the problem is formally determined. It is demonstrated that the final form of dynamical system and corresponding conservation law depends on the solution of the so-called potential equation. However, the structure of symmetry transformations, 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 pragmatic value is confirmed through several concrete examples.