Let $M$ be a compact manifold, and consider the space $H^1(S^1,M)$ of loops that are of sobolev class $W^{1,2}$. Under suitable conditions on a lagrangian $L$, i.e., non-degenerancy, quadratic at infinity and Tonelli, one can do Morse Homology on this space with the fuctional
$$\xi_L(q):=\int_{0}^{1}L(t,q(t),\dot q(t))dt$$
Namely I am interested in seeing that under the conditions on the lagrangian $L$ we will have that the critical points of $\xi$ are non-degenerate and have finite morse index.
I was wondering if anyone one knows a reference where this is done with some detail ? All I could find was Milnor's book on Morse theory but here $\xi$ is the energy functional and our critical points will be geodesics, therefore I was looking for something more general.
I was trying to prove these statements myself, but I am not sure if I should use local coordinates or try and use the levi-civita connection , and hence wanted to see what the standard treatment was.
Any insight is appreciated, thanks in advance.