Dirac brackets in constrained dynamics. (English) Zbl 0956.37049

Ibort, Alberto; de Leon, Manuel; Marrero, Juan C.; Martin de Diego, David

Fortschr. Phys. 47, No. 5, 459-492 (1999).

Summary: An unified geometric description of various Dirac brackets for regular and singular Lagrangians with holonomic or non-holonomic constraints is presented. Such an unified picture relies only on two simple physical arguments: “classical complementarity principle” and “principle of deterministic evolution”. The appropriate geometrization of these principles allows to construct a recursive constraint algorithm that eventually produces a maximal final state manifold where a well defined dynamics exists, naturally equipped with a Dirac bracket such that the dynamics is Hamiltonian with respect to it. A classification of constraints in first and second class as envisaged by Dirac emerges also naturally from this picture. The Dirac brackets constructed show explicitly the existence of classical anomalies for such Lagrangian theories since in general they do not verify Jacobi’s identity. Such features are discussed using a variety of examples.

MSC:

37J60	Nonholonomic dynamical systems
53D17	Poisson manifolds; Poisson groupoids and algebroids
70F25	Nonholonomic systems related to the dynamics of a system of particles
70H03	Lagrange’s equations
70H45	Constrained dynamics, Dirac’s theory of constraints

Keywords:

Dirac brackets; regular Lagrangians; singular Lagrangians; holonomic constraints; non-holonomic constraints; classical complementarity principle; principle of deterministic evolution; Jacobi’s identity