Generalized duality, Hamiltonian formalism and new brackets

TL;DR

This paper introduces a constraint-free Hamiltonian formalism for singular Lagrangian theories using a Clairaut-type approach, revealing connections to nonabelian gauge theories and many-time dynamics, and relating it to Dirac's constrained systems.

Contribution

It develops a generalized Hamiltonian formalism for singular Lagrangian theories that avoids constraints by employing a Clairaut-type transform, linking to gauge theories and many-time dynamics.

Findings

Formulates singular Lagrangian theories without constraints.

Establishes a Hamilton-like formalism with new antisymmetric brackets.

Shows equivalence to many-time classical dynamics and relates to Dirac's approach.

Abstract

It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dynamics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed (envelope/general) solution of the multidimensional Clairaut equation. The corresponding system of motion equations is equivalent to the Lagrange equations and has a linear algebraic subsystem for "unresolved" velocities. Then the equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. This is a "shortened" formalism, since it does not contain the "nondynamical" (degenerate) momenta at all, and therefore there is no notion of constraint. It is outlined that any classical degenerate Lagrangian theory (in its Clairaut-type Hamiltonian form) is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given.