Chapter 9.5, Problem 7P

Prove that a particle constrained to stay on a surface $f(x,y,z)=0,$ but subject to no other forces, moves along a geodesic of the surface. Hint: The potential energy $V$ is constant, since constraint forces are normal to the surface and so do no work on the particle. Use Hamilton’s principle and show that the problem of finding a geodesic and the problem of finding the path of the particle are identical mathematics problems.