The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics
W. Sarlet · Journal of Physics A Mathematical and General · 1982
Deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second-order ordinary equations the structure of Euler-Lagrange equations. The approach is based on a generalisation of previous studies on linear systems. The main result concerns a set of necessary and sufficient conditions for the existence of a multiplier, which contains an infinite set of algebraic equations, the coefficients of which can be used to derive necessary conditions involving only the given right-hand sides of the differential equations. An outline is given of interesting points for future studies, and an example is presented for which all multipliers are explicitly constructed.