Energy Balance for a Nonholonomic System with Nonlinear Constraints

This paper deals with mechanical systems subject to nonlinear nonholonomic constraints. The aspect that is mainly investigated is the formulation of an energy equation, which is deduced from the equations of motion, via the generalization of standard procedures pertaining to simpler cases (holonomic systems). We consider the equations of motion in two different forms (both present in the literature): the first method introduces the Lagrange multipliers, the second one is based on the selection of a certain number of independent velocities. The second procedure generalizes Voronec’s equations for linear kinematic constraints to the case of nonlinear kinematic constraints. The two kinds of equations of motion gives rise to two different ways of writing an energy-type equation, since in the second case only the independent velocities are used and restricted functions corresponding to the Lagrangian and to the Legendre transform consequently appear in the energy equation. We see that the discrepancy between energy and its reduced version disappears if and only if the explicit expressions of the constraints are homogeneous functions of degree 1 with respect to the independent velocities. This property has an effect to various aspects of the problem and produces certain benefits which place nonlinear nonholonomic system near to the linear ones. From the mathematical point of view, we prove that the homogeneity of the explicit functions is equivalent to the homogeneity of the given constraints with respect to the full set of velocities. This conclusion is in step with previous results, where the conservation of energy in nonlinear nonholonomic system is strictly connected with the homogeneity of the constraint functions. According to the present work, the known result is put into perspective together with other aspects of nonlinear nonholonomic systems, mainly investigating the unifying role exerted by the homogeneity property of the explicit constraint functions. A selection of cases and examples is discussed.