The inverse problem of Lagrangian dynamics for higher-order differential equations: a geometrical approach

Abstract

Crampin and co-workers (1981, 1984) showed that, under certain circumstances, there is a class of vector fields such that if X is in this class then there is a local Lagrangian G for which X satisfies the corresponding Euler-Lagrange equations. Such an approach leads to a geometrical version of the well known inverse problem of classical mechanics. The general purpose of this work is to make a contribution problem of classical mechanics. The general purpose of this work is to make a contribution to the renewed interest in the geometric formalism of Lagrangian theories involving higher-order derivatives (generalized theories). The authors examine the above inverse problem in the framework of tangent bundle geometry of order k, k< infinity , in such a way that some of Crampin's results are re-obtained for the standard situation k=1. They extend the theory to manifolds endowed with an integrable almost tangent structure of higher order.