# Coordinate-free proof of the hamiltonian character of the geodesic flow

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$

It is well known that:

The geodesics of $(M,g),$ i.e. the solutions of $\frac{D}{dt}\gamma=0,$ are integral curves for the hamiltonian vector field of $K:u\in TM\to \tfrac{1}{2}g(u,u)\in\mathbb{R}$ w.r.t. the canonical symplectic form.

Question Knowing how to show it using coordinates, I am wondering how to prove it in an intrinsic way.

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I have posted it even on MO, here: mathoverflow.net/questions/88624/… –  Giuseppe Tortorella Feb 16 '12 at 13:09

## 1 Answer

The MathOverflow version of this post has been answered there. There are indeed coordinate free proofs of the fact as the answers there demonstrate.

If someone prefers to have a more detailed answer here, feel free to summarize the MO answers. I just wanted to give this brief CW answer to help people find the answers and to remove this question from the unanswered list.

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