Geodesics as Hamiltonian flows

In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is developed in this article.

This can be interpreted in contact geometry as well using Reeb vector flow. Reeb vector flow applies more generally to the geodesic flow on Finsler manifolds.