Modified Hamilton formalism for fields

I. Danilenko<sup>ab

a</sup> Institute for Theoretical and Experimental Physics, Moscow, Russia
^b Moscow Institute of Physics and Technology (State University), Moscow, Russia

Abstract: In Hamiltonian mechanics, the equations of motion can be regarded as a condition on the vectors tangent to the solution: they should be null-vectors of the symplectic structure. The passage to the field theory is usually done by replacing the finite-dimensional configuration space with an infinite-dimensional one. We apply an alternative formalism in which the space–time is considered one worldsheet and its maps are studied. Instead of null-vectors of the symplectic $2$-form, null-polyvectors of a higher-rank form on a finite-dimensional manifold are introduced. The action in this case is an integral of a differential form over a surface in the phase space. Such a method for obtaining the Hamiltonian mechanics from the Lagrange function is a generalization of the Legendre transformation. The condition that the value of the action and its extremals are preserved naturally determines this procedure.

Keywords: Hamiltonian mechanics, field theory.

Received: 23.12.2012
Revised: 19.02.2013

DOI: 10.4213/tmf8460

 English version: Theoretical and Mathematical Physics, 2013, 176:2, 1067–1086

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