I am reading the introduction to Variational Integrators by Marsden and West and I see that the starting point to build the discrete Lagrangian is the theorem that states that for any two close enough points $q_0$, $q_1$ and time $h$ small enough, there exists a unique Lagrangian movement $q(t)$ with $q(0)=q_0$ and $q(h)=q_1$. This is remindful of the existence of local geodesics, just we do not have a metric here, since the context is that of abstract differentiable manifolds.
They give Marsden and Ratiu as a reference for the proof of this theorem, but I am unable to find this exact statement. Maybe it is hidden into a more general fact. I would appreciate if you point out the particular page of this or another text, or provide a proof of this seemingly elementary fact.
Feel free to ask for clarifications.