Time reversibility and energy conservation for Lagrangian systems with nonlinear nonholonomic constraints

When is a nonholonomic Lagrangian system time-reversible? We prove that a simple sufficient condition is that (skipping over some minor technicalities) both the Lagrangian $L(t,q,\dot q)$ and the set of the triples $(t,q,\dot q)$ that satisfy the constraints are invariant by exchange of $(t,\dot q)$ into $(-t,-\dot q)$. Another question is: when is energy conserved in a nonholonomic autonomous Lagrangian system? A likewise easy sufficient condition is that the set of the couples $(q,\dot q)$ satisfying the constraints is a cone with respect to~$\dot q$ (meaning that if $(q,\dot q)$ is admissible then $(q,r\dot q)$ is admissible too for all $r\ge0$). Time-reversibility and energy conservation are independent properties, in the sense that neither one implies the other. Both properties hold at the same time for any autonomous system with a ``natural'' Lagrangian and with constraints that are homogenous in $\dot q$.