Given an affine manifold $(M,\nabla)$, the geodesic equation $\ddot{x}^j+\dot{x}^k \dot{x}^l\Gamma_{kl}^j=0$ completely characterizes the geodesics on the manifold. This is often called the Euler-Lagrange equation. I was wondering what the connection between geodesics and Lagrangian mechanics are. Given a mechanical system, its solutions can be found by solving the resulting Euler-Lagrange equation. Is there some affine connection $\nabla$ we can equip the configuration space $C$ so that all solutions are hence geodesics on the "manifold" $(C,\nabla)$? Reading this paper, it seems one can construct a Riemannian metric from the kinetic energy. However, it seems this really only works when there's no potential in the system, which I find kinda odd since what about the case where there is a potential.
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