How to show that geodesics exist for all of time in a compact manifold? Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form $\omega_{std}$ to define a symplectic form $\omega_g$ on $TM. $
Assuming the following theorem:

The geodesics in $M$ are the images by the standard projection $$\pi: TM \to M$$ of the integral curves of the vector field $X_H$ where $H$ is the norm function squared in TM defined by the metric $g$

I want to show that the geodesics exist for all of time. 
Please help by giving some ideas!
 A: The short-time existence & uniqueness theorem for ODE's contains estimates for the length of the interval on which the solution exists in terms of the coefficients and their derivatives and the size of the chart you must remain within.
This implies that an ODE on a compact manifold (not quite your situation) admits a solution for all time: Cover by finitely many half-charts, get the bounds in terms of the coefficients and the sizes of the charts, and get a global $\epsilon > 0$ such that the ODE can always be solved for time $\epsilon$. Then it can be solved for all time just by "moving forward in the interval" and repeatedly solving for $\epsilon$ more time starting from the end of your previous solution.
Your situation is slightly different, because your manifold, $TM$, is not compact. But you can prove that your integral curves always stay the same length because they remain on a level set for $H$. So in fact if you start in the interior of a compact $B_R(TM)$ (consisting of the closed $R$ ball around $0$ in each tangent space) then you will remain there (and also remain at a fixed distance from the boundary of this). So the same proof as in the compact case works.
