Metric nonlinear connections
  For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange structure. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonic nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. The metric tensor of the Lagrange space determines the symmetric part of the nonlinear connection, while the symplectic structure of the Lagrange space determines the skew-symmetric part of the nonlinear connection. 
Ioan Bucataru