Cubic invariants of one-dimensional Lagrangian systems

Abstract

In this paper we consider conservation laws of the third degree with respect to ẋ of the one-dimensional time-dependent Lagrangian systems ẍ = - ∂Π/∂x. The analysis is based on the Noetherian approach. It is shown that the existence of conservation laws, as well as their structure depend on the solution of a system of first-order partial differential equations - so-called generalized Killing's equations. It is demonstrated that due to specific structure of the generators of infinitesimal transformations a rather general algorithm for derivation of cubic invariants could be established. Several types of potential Π(t,x) which admit the existence of cubic invariants are determined.