Inverse Problem For Lagrangian Mechanics - Background and Statement of The Problem

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.