sentences:  A notable advance in this field was a 1941 paper by the American mathematician Jesse Douglas, in which he provided necessary and sufficient conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the German physicist Hermann von Helmholtz.The usual set-up of Lagrangian mechanics on "n"-dimensional Euclidean space R"n" is as follows.Consider a differentiable path "u"