Differential-geometric aspects of a nonholonomic Dirac mechanics: Lessons of a model quadratic in velocities

Faddeev and Vershik proposed the Hamiltonian and Lagrangian formulations of constrained mechanical systems that are invariant from the differential geometry standpoint. In both formulations, the description is based on a nondegenerate symplectic $2$-form defined on a cotangent bundle $T^*Q$ (in the Hamiltonian formulation) or on a tangent bundle $TQ$ (in the Lagrangian formulation), and constraints are sets of functions in involution on these manifolds. We demonstrate that this technique does not allow “invariantization” of the Dirac procedure of constraint “proliferation.” We show this in an example of a typical quantum field model in which the original Lagrange function is a quadratic form in velocities with a degenerate coefficient matrix. We postulate that the initial phase space is a manifold where all arguments of the action functional including the Lagrange multipliers are defined. The Lagrange multipliers can then be naturally interpreted physically as velocities (in the Hamiltonian formulation) or momenta (in the Lagrangian formulation) related to “nonphysical” degrees of freedom. A quasisymplectic $2$-form invariantly defined on such a manifold is degenerate. We propose new differential-geometric structures that allow formulating the Dirac procedure invariantly.