Systems of second order differential equations as auto-parallel equations

Abstract

Nonlinear connections and associated horizontal Berwald-type connections whose auto-parallel equations coincide with an arbitrary system of second order differential equations analytic in the velocities are constructed explicitly. For autonomous Lagrangian systems the nonlinear connection can be chosen in such a way that the horizontal derivative of the conserved Hamiltonian vanishes identically. For natural Lagrangian systems this connection is equivalent to the Levi-Civita connection of the associated Jacobi metric. The connection constructed from the Euler-Lagrange equations of a particle in an electromagnetic and gravitational background naturally provides a unified geometrical description of classical fundamental interactions.