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 Zh. Mat. Fiz. Anal. Geom., 2014, Volume 10, Number 2, Pages 189–220 (Mi jmag588)

Generalized Duality, Hamiltonian Formalism and New Brackets

S. Duplij

Theory Group, Nuclear Physics Laboratory, V. N. Karazin Kharkiv National University, 4 Svoboda Sq., Kharkiv 61022, Ukraine

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 equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given.

Key words and phrases: Dirac constraints, nonabelian gauge theory, degenerate Lagrangian, Hessian, Legendre transform, multidimensional Clairaut equation, gauge freedom, Poisson bracket, many-time dynamics.

DOI: https://doi.org/10.15407/mag10.02.189

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Document Type: Article
MSC: 37J05, 44A15, 49K20, 70H45
Revised: 16.07.2013
Language: English

Citation: S. Duplij, “Generalized Duality, Hamiltonian Formalism and New Brackets”, Zh. Mat. Fiz. Anal. Geom., 10:2 (2014), 189–220

Citation in format AMSBIB
\Bibitem{Dup14} \by S.~Duplij \paper Generalized Duality, Hamiltonian Formalism and New Brackets \jour Zh. Mat. Fiz. Anal. Geom. \yr 2014 \vol 10 \issue 2 \pages 189--220 \mathnet{http://mi.mathnet.ru/jmag588} \crossref{https://doi.org/10.15407/mag10.02.189} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3236967} 

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This publication is cited in the following articles:
1. Walker M.L., Duplij S., “Cho-Duan-Ge Decomposition of Qcd in the Constraintless Clairaut-Type Formalism”, Phys. Rev. D, 91:6 (2015), 064022
2. Duplij S., “Formulation of Singular Theories in a Partial Hamiltonian Formalism Using a New Bracket and Multi-Time Dynamics”, Int. J. Geom. Methods Mod. Phys., 12:1 (2015), 1550001
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