Inverse problem for Lagrangian mechanics

Remarks

Equivalently, one asks for an invertible matrix of multipliers $g_{ij}(t, q, \dot q)$ such that $\sum_j g_{ij} (\ddot q_j - F_j)$ is the Euler–Lagrange expression of some $L$; then $g_{ij} = \partial^2 L / \partial \dot q_i \partial \dot q_j$. The conditions that such a matrix must satisfy are a system of partial differential equations known as the Helmholtz conditions. For $n = 1$, every such equation has a Lagrangian locally. Jesse Douglas treated the case $n = 2$ in 1941 and sorted the systems into classes according to whether a Lagrangian exists and how many there are. For $n \ge 3$ no complete answer is known. The answer is not unique when it exists. For $n = 1$ and $\ddot q = 0$, both $L = \frac{1}{2} \dot q^2$ and $L = e^{\dot q}$ work, since the Euler–Lagrange equation of $e^{\dot q}$ is $e^{\dot q} \ddot q = 0$, that is $\ddot q = 0$.