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

When is a nonholonomic Lagrangian system time-reversible? We prove that a s imple sufficient condition is that (skipping over some minor technicalities ) both the Lagrangian L(t,q,(g) over dot) and the set of the triples (t,q,( q) over dot) that satisfy the constraints are invariant by exchange of (t,( q) over dot) into (-t, -(q) over dot). Another question is: when is energy conserved in a nonholonomic autonomous Lagrangian system? A likewise easy s ufficient condition is that the set of the couples (q, (q) over dot) satisf ying the constraints is a cone with respect to (q) over dot (meaning that i f (q, (q) over dot) is admissible then (q,r(q) over dot) is admissible too for all r greater than or equal to 0). Time-reversibility and energy conser vation are independent properties, in the sense that none of them implies t he other. Both properties hold at the same time for any autonomous system w ith a "natural" Lagrangian and with constraints that are homogenous in (q) over dot.