On the geometric formulation of Hamiltonian dynamics

Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories can be identified as geodesics of the underlying manifold. We show how these geometric structures can be derived; specifically, we construct them explicitly for configuration and phase spaces of Hamiltonian systems. We demonstrate how the correspondence between geometry and dynamics can be applied to study the conserved quantities of a dynamical system. Lastly, we demonstrate how the mean-curvature of the energy level-sets in phase-space might be correlated with strongly chaotic behavior.