The nonholonomic variational principle

Abstract

A variational principle for mechanical systems and fields subject to nonholonomic constraints is found, providing Chetaev-reduced equations as equations for extremals. Investigating nonholonomic variations of the Chetaev type and their properties, we develop foundations of the calculus of variations on constraint manifolds, modelled as fibred submanifolds in jet bundles. This setting is appropriate to study general first-order ‘nonlinear nonitegrable constraints’ that locally are given by a system of first-order ordinary or partial differential equations. We obtain an invariant constrained first variation formula and constrained Euler–Lagrange equations both in intrinsic and coordinate forms, and show that the equations are the same as Chetaev equations ‘without Lagrange multipliers’, introduced recently by other methods. We pay attention to two possible settings: first, when the constrained system arises from an unconstrained Lagrangian system defined in a neighbourhood of the constraint, and second, more generally, when an ‘internal’ constrained system on the constraint manifold is given. In the latter case a corresponding unconstrained system need not be a Lagrangian, nor even exist. We also study in detail an important particular case: nonholonomic constraints that can be alternatively modelled by means of (co)distributions in the total space of the fibred manifold; in nonholonomic mechanics this happens whenever constraints affine in velocities are considered. It becomes clear that (and why) if the distribution is completely integrable (= the constraints are semiholonomic), the principle of virtual displacements holds and can be used to obtain the constrained first variational formula by a more or less standard procedure, traditionally used when unconstrained or holonomic systems are concerned. If, however, the constraint is nonintegrable, no significant simplifications are available. Among others, some properties of nonholonomic systems are clarified that without a deeper insight seem rather mysterious.