Nonlinear connections for nonconservative mechanical systems

The geometry of a nonconservative mechanical system is determined by its associated semispray and the corresponding nonlinear connection. The semispray is uniquely determined by the symplectic structure and the energy of the corresponding Lagrange space and the external force field. We prove that the corresponding nonlinear connection is uniquely determined by its compatibility with the metric tensor and the symplectic structure of the Lagrange space. We study the variation of the energy and Lagrangian functions along the evolution curves and the horizontal curves and give conditions by which these variations vanish. We provide examples of mechanical systems which are dissipative and for which the evolution nonlinear connection is either metric or symplectic.