Generalized duality, Hamiltonian formalism and new brackets

Details Generalized duality, Hamiltonian formalism and new brackets Generalized duality, Hamiltonian formalism and new brackets

2010-02-08

arxiv.org/abs/1002.1565v4

Physics

Mathematical Physics

21 pages, amsart, added: 1) Section 6 on analogy with a kind of nonabelian gauge theory, where "nonabelianity" is "classical"

Scientific paper

We show 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; 3) coincides with the many-time classical dynamics. A generalization of the Legendre transform to the case when the Hessian is zero is done by using the mixed (envelope/general) solution of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the Lagrange equations and has a linear algebraic subsystem for "unresolved" velocities. Then the equations of motion are written in the Hamiltonian form by introducing a new bracket. This is a "shortened" formalism, since initially it does not contain "nondynamical" (degenerate) momenta at all, and therefore there is no notion of constraint. It is shown that any classical singular Lagrangian theory in its Clairaut-type Hamiltonian form is equivalent to the many-time classical dynamics.

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