Let $M$ be a compact smooth manifold and consider $L:S^1\times TM\rightarrow \mathbb{R}$ a smooth $1$-periodic Lagrangian. If we assume that $d_{vv} L(t,q,v)\geq l_0I$ for some $l_0>0$ then one can show that the morse index of the action functional at a curve $q(t):S^1\rightarrow M$ of type $W^{1,2}.$
$$\mathcal{E}_L(q):=\int_0^1L(t,q(t),\dot q(t))$$
is finite.
Now if we consider $L$ to be the energy function, that is $L(t,q,v)=\frac{1}{2}|v|^2$, Milnor proves in his book that the Morse index at a critical point $x$ actually equals the number of conjugate points along the critical point $x(t). $
Now I wonder what happens in the case of a more general Lagrangian $L_t$ satisfying the condition $d_{vv}L(t,q,v)\geq l_0I$. Is there a way in which we can compute the second differential so that we can get an analogous result ?
Any insight is appreciated, thanks in advance.