.  If $t>0$, the function $A_t~: M\times M\rightarrow {\mathbb {R}}$ is defined by: $$A_t(q_0, q_0)=\inf_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds=\min_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds$$ where the infimum is taken on the set of $C^0$ curves $\gamma~: [0, t]\rightarrow M$ such that $\gamma (0)=q_0$ and $\gamma (t)=q_0$.  0.  