The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket (Q2737921)

There are two different approaches to Lagrangian systems subjected to nonholonomic constraints: nonholonomic mechanics, based on the d'Alembert principle, and vakonomic mechanics, which is strictly variational. The aim of this paper is to study the equations of motion of vakonomic mechanics, in the framework of singular Lagrangian theories, from a geometric viewpoint. NEWLINENEWLINENEWLINEMore precisely, let \(Q\) be the configuration manifold. It is assumed that the Lagrangian function is of natural type, that is, \(L=T-U\), where \(T\) is the kinetic energy derived from a Riemannian metric on the base manifold, and U is the potential energy. It is also assumed that the constraint submanifold of the tangent manifold \(TQ\) is globally defined by the vanishing of a family of constraint functions, which are supposed to be linear in the velocities. Then, the configuration space \(Q\) is enlarged to \(P\) by introducing a family of Lagrange multipliers as new coordinates. The constraint algorithm is applied to the extended Lagrangian function in \(TP\) and the dynamics is described (Section IV of the paper), showing that the algorithm stabilizes at the second step, i.e., there are only secondary constraints. NEWLINENEWLINENEWLINENext, a Hamiltonian formalism on the cotangent manifold of \(P\) is discussed in Section V. In particular, it is easily proven that the extended Lagrangian is almost regular, and then the equivalence between the both formalisms, Lagrangian and Hamiltonian ones, is discussed. The second order differential equation problem is studied in Section VI. In Section VII it is shown that all the constraints are second class according to Dirac's terminology, and a Dirac bracket giving the evolution of the observables is constructed. As an example, the vertical rolling disk is analyzed. The paper finishes with an study of the situation when the constraints are not globally defined in the original phase space \(TQ\).