On the Global Version of Euler-lagrange Equations

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version. Submitted to: J. Phys. A: Math. Gen. PACS numbers: 45.10.Na 02.40.Yy The introduction of numbers as coordinates. . . is an act of violence Hermann Weyl An increasing attention has been recently paid to coordinate free formulations of motion equations in Classical Mechanics (see for instance [1, 2] and references therein). In this work we write down intrinsic Euler-Lagrange equations and show the appearance of a torsion term. Furthermore, we shall see that this term should also be present in the horizontal Lagrange-Poincaré equations considered in [1, 2], if the torsion of the chosen derivative does not vanish. It is worth noticing that covariant derivatives with non vanishing torsion naturally arise in several branches of Physics; namely, dynamics with nonholonomic constraints [3, 4], E. Cartan’s Theory of Gravity (see for instance [5]) and modern string theories (see for example [6]), among others. Let us consider a physical system with configuration manifold Q and Lagrangian L(q, q̇) : TQ → R (for this geometrical setting see for instance [7]). If a coordinate free characterization of the Euler-Lagrange Equations associated to the system is required a covariant derivative D must be introduced on TQ, for ∂L ∂q is involved (see for instance [8]). Once such D is chosen, DL Dq is defined in the standard way DL Dq (q0, q̇0) = ∂ ∂λ ∣