Generalized Duality, Hamiltonian Formalism and New Brackets

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.

Cho-Duan-Ge decomposition of QCD in the constraintless Clairaut-type formalism

“…The difference between our new brackets {, } new and Dirac brackets is clarified in app. B of [6]. If one introduces additional "nonphysical" momenta p α (equation (B1) in [6] or sec.…”

“…B of [6]. If one introduces additional "nonphysical" momenta p α (equation (B1) in [6] or sec. 5 of [7]) corresponding to the "nonphysical" coordinates q α , then the new bracket in the fully extended phase space becomes the Dirac bracket.…”

“…Eq. (41) can therefore be considered to be a new shortened version of quantization for singular systems, as described in the conclusions of [6] and [7].…”

“…A constraintless generalization of the Hamiltonian formalism based on a Clairaut-type formulation was recently put forward by one of the authors [6,7]. It generalises the standard Hamiltonian formalism to include Hessians with zero determinant, providing a rigorous treatment of the non-physical degrees of freedom in the derivation of EOMs and the quantum commutation relations.…”

Variational problem for Hamiltonian system on so(k, m) Lie–Poisson manifold and dynamics of semiclassical spin

“…Hamiltonian systems on Poisson manifolds naturally arise during analysis of many classical problems [2][3][4] and in modern extensions and applications of Hamiltonian formalism [5][6][7][8][10][11][12]. Numerous examples of dynamics on nontrivial Poisson manifolds can be obtained applying the Dirac procedure for analysis of constrained systems to singular Lagrangian theories [5,9,12,13,16].…”