New article "Polysymplectic Hamiltonian field theory"

My new article "Polysymplectic Hamiltonian field theory" arXiv: 1505.01444

Abstract

Applied to field theory, the familiar symplectic technique leads to instantaneous Hamiltonian formalism on an infinite-dimensional phase space. A true Hamiltonian partner of first order Lagrangian theory on fibre bundles Y->X is covariant Hamiltonian formalism in different variants, where momenta correspond to derivatives of fields relative to all coordinates on X. We follow polysymplectic (PS) Hamiltonian formalism on a Legendre bundle over Y provided with a polysymplectic TX-valued form. If X=R, this is a case of time-dependent non-relativistic mechanics. PS Hamiltonian formalism is equivalent to the Lagrangian one if Lagrangians are hyperregular. A non-regular Lagrangian however leads to constraints and requires a set of associated Hamiltonians. We state comprehensive relations between Lagrangian and PS Hamiltonian theories in a case of semiregular and almost regular Lagrangians. Quadratic Lagrangian and PS Hamiltonian systems, e.g. Yang - Mills gauge theory are studied in detail. Quantum PS Hamiltonian field theory can be developed in the frameworks both of familiar functional integral quantization and quantization of the PS bracket.

Contents

• First order Lagrangian formalism on fibre bundles
• Cartan and Hamilton - De Donder equations
• Polysymplectic structure
• PS bracket
• Hamiltonian forms
• Covariant Hamilton equations
• Hamiltonian time-dependent mechanics
• Iso-PS structure
• Associated Hamiltonian and Lagrangian systems
• Lagrangian and Hamiltonian conservation laws
• Lagrangian and Hamiltonian Jacobi fields
• Quadratic Lagrangian and Hamiltonian systems
• PS Hamiltonian gauge theory
• Affine Lagrangian and Hamiltonian systems
• Functional integral quantization
• Algebraic quantization. Quantum PS bracket

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Conference "Geometry of Jets and Fields"

International Conference Geometry of Jets and Fields (10-16 May 2015, Bedlewo, Poland) on the 60th birthday of Janusz Grabowski

Plenary Talks (Abstracts and slides) (#)

My Plenary Talk: Noether theorems in a general setting. Reducible graded Lagrangians (#)

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My new "Handbook of Integrable Hamiltonian Systems"

My new book: G. Sardanashvily, “Handbook of Integrable Hamiltonian Systems” (URSS, 2015) has been published.

This book provides comprehensive exposition of completely integrable, partially integrable and superintegrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. In particular, this is the case of non-autonomous integrable Hamiltonian systems and integrable systems with time-dependent parameters. The fundamental Liouville – Minuer – Arnold, Poincare – Lyapunov – Nekhoroshev, and Mishchenko – Fomenko theorems and their generalizations are present in details. Global action-angle coordinate systems, including the Kepler one, are analyzed. Geometric quantization of integrable Hamiltonian systems with respect to action-angle variables is developed, and classical and quantum Berry phase phenomenon in completely integrable systems is described. The book addresses to a wide audience of theoreticians and mathematicians of undergraduate, post-graduate and researcher levels. It aims to be a guide to advanced geometric methods in classical and quantum Hamiltonian mechanics. For the convenience of the reader, a number of relevant mathematical topics are compiled in Appendixes

Preface, Contents, Introduction

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