Riemannian metric for an autonomous second-order ODE

Given an autonomous second-order ODE

u^{″} = ϕ (u, u_{1}),

with $u_{1} = u^{'}$ , we define a Riemannian metric on an open subset of the first-order jet bundle $J^{1} (R, R)$ :

M = {(x, u, u_{1}) \in U : u_{1} \neq 0},

with

g = (1 + u_{1}^{2}) d x^{2} - 2 u_{1} d x d u + (1 + \frac{ϕ^{2}}{u_{1}^{2}}) d u^{2} - 2 \frac{ϕ}{u_{1}} d u d u_{1} + d u_{1}^{2} .

The associated vector field $A = \partial_{x} + u_{1} \partial_{u} + ϕ \partial_{u_{1}}$ has unit length and is a geodesic vector field. The orthonormal frame is

e_{1} = A, e_{2} = \partial_{u} + \frac{ϕ}{u_{1}} \partial_{u_{1}}, e_{3} = \partial_{u_{1}},

with dual coframe

ω^{1} = d x, ω^{2} = - u_{1} d x + d u, ω^{3} = - \frac{ϕ}{u_{1}} d u + d u_{1} .

Geodesics of $(M, g)$ correspond to prolonged solutions: if $f$ solves the ODE, then $j^{1} f$ is a geodesic. The converse holds when $(ϕ / u_{1})_{u_{1}} \neq 0$ .

Reference: @panalvarez2024autonomous