On Euler's Theorem for the Motion Along Geodesies

摘要

We reviewed briefly the problem of the free motion of a particle along geodesies of a smooth surface under no forces according to the geometrical and dynamical points of view. We prove for Euler's the-orem-the coincidence of the purely analytical geometrical approach of L.P. Eisenhart and the dynamical approach of E.T Whittaker using the Lagrangian equations of motion and Christoffel's symbols. We intergrated the two ordinary differential equations describing the motion in the geodesic trajectory with respect to the two curvilinear coordinates u,v.