In this paper, we use the notion of Grifone’s connection over a tangent bundle in order to construct a connection over a sub-bundle. Then we characterize the solutions of non-holonomic Lagrangian mechanics and show that the geodesics of the connection constructed on the sub-bundle are the solutions of the non-holonomic Euler-Lagrange system. Finally, we will prove that the Hamiltonian associated to Lagrangian function is constant along the horizontal curves.

connections, distributions, Lagrangian mechanics, fibre bundle.