Canonical transformations of local functionals and sh-Lie structures

Abstract

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms.We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras.

Symmetry reduction of sh-Lie structures and of local functionals

“…Now notice that X = r y (totX 1 ∧ totX 2 ) is a Ginvariant 2-chain, and that ̺ X (dx ∧ dy ∧ dz) = r 2 dr, where ν = dx ∧ dy ∧ dz is G-invariant and ν = r 2 dr. We conclude that there exists of a reduced sh-Lie structure. Now suppose that Remark In [1] reduction of local functionals and sh-Lie structures was studied for the case when the Lie group G acts only on the fibers, i.e., when the induced action on the base manifold M is just the identity map. In the current paper we assumed that the action is transversal to the fibers.…”

“…For example one can consider a reduction from π to π under a Lie group action as was done in this paper. Then another reduction on Ω * ,0 c (J ∞ E) may be obtained when another Lie group acts on the fibers of π as in [1].…”

“…In [1] ideas related to canonical transformations of these structures were studied as well as ideas of reduction. In this paper we investigate how the ideas of symmetry reduction as in [3] can be applied to obtain a reduction of sh-Lie structures and of functionals.…”

Canonical transformations and Hamiltonian evolutionary systems

“…It would be interesting to study the sh-Lie algebra structure and reduction as was done in [1] and [3]. However we leave this for a possible future work.…”

“…This terminology was used in [1] and is in anology to the one used in [7] for the case of symplectic manifolds. In [1] canonical transformations were studied in the case the Poisson bracket is defined by a differential operator of order zero. In this paper we study canonical transofrmations in the case of higher order differential operators.…”