Energy foliation

On the Riemannian manifold associated to an autonomous second-order ODE $u^{″} = ϕ (u, u_{1})$ , the distribution $D = span {e_{1}, e_{2}}$ is involutive:

[e_{1}, e_{2}] = - \frac{ϕ}{u_{1}} e_{2} .

By Frobenius' theorem, it defines a foliation $E$ , called the energy foliation. Its leaves are locally given by level sets of a function $E$ satisfying

d E = μ ω^{3} ⟺ E_{u} + \frac{ϕ}{u_{1}} E_{u_{1}} = 0.

Properties:

The energy foliation is a minimal foliation of $(M, g)$ .
For ODEs arising from Lagrangian mechanics, the leaves correspond to constant-energy surfaces, justifying the name.
Every function $E$ defining $E$ yields an autonomous Lagrangian via $L (u, u_{1}) = u_{1} \int E (u, u_{1}) / u_{1}^{2} d u_{1}$ , solving the inverse problem of the calculus of variations for these ODEs.

Reference: @panalvarez2024autonomous