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,q̇) and the set of the triples (t,q,q̇) that satisfy the constraints are invari exchange of (t,q̇) into (−t,−q̇). Another question is: when is energy conserved i autonomous Lagrangian system? A likewise easy sufficient condition is that the set of the couples (q,q̇) satisfying the constraints is a cone with respect to q̇ if (q,q̇) is admissible then (q,rq̇) is admissible too for all r ≥ 0). Time-rev energy conservation are independent properties, in the sense that none of them 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 q̇.