Inverse problem for Lagrangian mechanics

Via wikipedia.

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.

Douglas' Theorem and the Helmholtz Conditions

To simplify the notation, let

v^{i} = {\dot{u}}^{i}

and define a collection of $n^{2}$ functions $Φ_{j}^{i}$ by

Φ_{j}^{i} = \frac{1}{2} \frac{d}{d t} \frac{\partial f^{i}}{\partial v^{j}} - \frac{\partial f^{i}}{\partial u^{j}} - \frac{1}{4} \frac{\partial f^{i}}{\partial v^{k}} \frac{\partial f^{k}}{\partial v^{j}} .

Theorem. (Douglas 1941) There exists a Lagrangian $L : [0, T] \times T M \to R$ such that the equations $(E)$ are its Euler–Lagrange equations if and only if there exists a non-singular symmetric matrix $g$ with entries $g_{i j}$ depending on both $u$ and $v$ satisfying the following three Helmholtz conditions:

$g Φ = (g Φ)^{⊤}$ (H1)
$g Φ = (g Φ)^{⊤}, (H1)$

\frac{d g_{i j}}{d t} + \frac{1}{2} \frac{\partial f^{k}}{\partial v^{i}} g_{k j} + \frac{1}{2} \frac{\partial f^{k}}{\partial v^{j}} g_{k i} = 0 for 1 \leq i, j \leq n, (H2)

\frac{\partial g_{i j}}{\partial v^{k}} = \frac{\partial g_{i k}}{\partial v^{j}} for 1 \leq i, j, k \leq n . (H3)

Surely, this gives rise to the Helmholtz operator.