Helmholtz condition
In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.
There has been a great deal of activity in the study of this problem since the early 20th century. 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 Rn is as follows. Consider a differentiable path u : [0, T] → Rn. The action of the path u, denoted S(u), is given by
where L is a function of time, position and velocity known as the Lagrangian. The principle of least action states that, given an initial state x0 and a final state x1 in Rn, the trajectory that the system determined by L will actually follow must be a minimizer of the action functional S satisfying the boundary conditions u(0) = x0, u(T) = x1. Furthermore, the critical points (and hence minimizers) of S must satisfy the Euler–Lagrange equations for S:
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