We discuss in this article the canonical structure of classical field theory in finite dimensions within the pataplectic Hamiltonian formulation, where we put forward the role of Legendre correspondence. We define the generalized Poisson 𝔭-brackets which are the analogs of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of 𝔭-brackets. As illustration of our formalism we present three examples: the interacting scalar fields, conformal string theory and the electromagnetic field.
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) De DonderWeyl (DDW) one. One of the main points is the fact that the Legendre transform in the DDW approach is replaced by a Legendre correspondence in the LP theory (this correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).e-print archive: http://lanl.arXiv.org/abs/math-ph/0401046 566 COVARIANT HAMILTONIAN FORMALISM . . .
This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable (n − 1)-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law {p, q} = 1 or the law dF/dt = {H, F }. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that -in contrast with the de Donder-Weyl theory -the two definitions coincides in the Lepage-Dedecker theory.e-print archive:
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