  Using the works of Ma\~n\'e \cite{Ma} and Paternain \cite{Pat} we study the
distribution of geodesic arcs with respect to equilibrium states of the
geodesic flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$
Riemannian metric. We prove large deviations lower and upper bounds and a
contraction principle for the geodesic flow in the space of probability
measures of the unit tangent bundle. We deduce a way of approximating
equilibrium states for continuous potentials.
