In section 1.1 under the subtitle system of classical particles with potential, the authors claim that "for a system of classical particles with rigid constraints, the configuration space is a Riemannian manifold X with Riemannian structure given by twice the kinetic energy."

I don't quite how the configuration space can be given by a Riemannian manifold, as it is more naturally viewed as a symplectic manifold and there appears to be no natural Riemannian structure on a symplectic manifold. Also the relation between the Riemannian structure and the kinetic energy also eludes me. The best interpretation I can think of is to impose a Riemannian structure on the cotangent bundle via Legendre transform, or the specification of a Lagrangian function. But this is not explcitly given.