Dynamic convexity for natural thermostatted systems

Natural thermostatted systems are mechanical systems whose Lagrangian is the difference of a kinetic and a potential energy, subjected to the nonholonomic constraint of a constant kinetic energy. When any two points of the configuration space are joined by a thermostatted motion, we say that the system is dynamically convex. A thermostatted charged particle on the plane with a constant electric field is not a dynamically convex system. We prove a general sufficient condition for dynamic convexity, from which whole classes of examples are easily constructed.