Background and Statement of The Problem
The usual set-up of Lagrangian mechanics on n-dimensional Euclidean space Rn is as follows. Consider a differentiable path u : → 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 in Rn, the trajectory that the system determined by L will actually follow must be a minimizer of the action functional S satisfying the initial condition u(0) = x0. Furthermore, the critical points (and hence minimizers) of S must satisfy the Euler–Lagrange equations for S:
where the upper indices i denote the components of u = (u1, ..., un).
In the classical case
the Euler–Lagrange equations are the second-order ordinary differential equations better known as Newton's laws of motion:
The inverse problem of Lagrangian mechanics is as follows: given a system of second-order ordinary differential equations
that holds for times 0 ≤ t ≤ T, does there exist a Lagrangian L : × Rn × Rn → R for which these ordinary differential equations (E) are the Euler–Lagrange equations? In general, this problem is posed not on Euclidean space Rn, but on an n-dimensional manifold M, and the Lagrangian is a function L : × TM → R, where TM denotes the tangent bundle of M.
Read more about this topic: Inverse Problem For Lagrangian Mechanics
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