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.
