A particle is constrained to move over a smooth fixed surface under no forces other than the force of constraint. By using Hamilton's principle and energy conservation, show that the path of the particle must be a geodesic of the surface. (The term geodesic has been extended here to mean those paths that make the length functional stationary).

This result has a counterpart in the theory of general relativity, where the concept of force does not exist and particles move along the geodesics of a curved space-time.