Modified Hamilton formalism for fields

摘要

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.