The God has created a man in order that he creates that the God fails to do

Thursday, 28 May 2015

New article "Polysymplectic Hamiltonian field theory"

My new article "Polysymplectic Hamiltonian field theoryarXiv: 1505.01444


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.

  • 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

Saturday, 16 May 2015

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 (#)

Friday, 8 May 2015

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