Non-holonomic constrained systems as implicit differential equations

Abstract

Non-holonomic constraints, both in the Lagragian and Hamiltonian formalism, are discussed from the geometrical viewpoint of implicit differential equations. A precise statement of both problems is presented remarking the similarities and differences with other classical problems with constraints. In our discussion, apart from a constraint submanifold, a field of permitted directions and a system of reaction forces are given, the later being in principle unrelated to the constraint submanifold. An implicit differential equation is associated to a non-holonomic problem using the Tulczyjew's geometrical description of the Legendre transformation. The integrable part of this implicit differential equation is extracted using an adapted version of the integrability algorithm. Moreover, sufficient conditions are found that guarantees the compatibility of the non-holonomic problem, i.e., that assures that the integrability algorithm stops at first step, and moreover it implies the existence of a vector field whose integral curves are the solutions to the problem. In addition this vector field turns out to be a second order differential equation. These compatibility conditions are shown to include as particular cases many others obtained previously by other authors. Several examples and further lines of development of the subject are also discussed.