I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where
$$H = \frac{1}{2}g^{ij}p_i p_j$$
but I am stuck. Could somebody show me how to complete the calculation, or where I've made a mistake? Cheers!
I know that vector field for geodesic flow is
$$X = p^i \partial/\partial x^i - \Gamma^i_{jk}p^j p^k \partial /\partial p^i$$
so I must verify that
$$X \ \lrcorner\ \omega = - dH$$
where $\omega = dp_i \wedge dx^i$. It's easy to check that
$$X \ \lrcorner\ \omega = - p^i dp_i - \Gamma^l_{jk}g_{li}p^j p^k dx^i$$
$$-dH = -g^{ij}p_j dp_i -\frac{1}{2}\partial_i g^{jk}p_j p_k dx^i$$
Here I am stuck. Using the explicit formula for the Christoffel symbols in terms of the metric doesn't seem to work! Have I done something wrong, or am I missing something?
N.B. I'm aware of solutions that do not use coordinates, but I'd like to understand this one which does!