It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations. In thumbnail form, the construction is as follows. Both M and Φ Φ are Riemannian manifolds; the below is written for Φ Φ , the same can be done for M . The cotangent bundle T ∗ ∗ Φ Φ , supplied with coordinate charts, can always be locally trivialized, i.e.