Mark J. Gotay

A Universal Reduction Procedure for Hamiltonian Group Actions

We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the Marsden-Weinstein reduction is well-defined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and Marsden-Weinstein reduction coincide. As an example, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.

A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations: I. Covariant Hamiltonian Formalism

Several recent results on the Hamiltonian formalism in the calculus of variations are presented. In particular, I propose a new candidate for the covariant phase space and show that it carries a canonical multisymplectic structure. Corresponding covariant Legendre transformations are constructed; while not necessarily unique, the class of all such is completely characterized. A suitable notion of regularity is also defined. These results comprise the foundation of a truly Hamiltonian framework for the calculus of variations in general, and enable one to deal directly with higher order Lagrangians as well as multiple integrals in much the same way as one treats ordinary mechanics. The key ingredient in this work is a generalization of Kijowski and Szczyrbas notion of “multiphase space.”

Constraints, reduction, and quantization

Theorems are proved that establish the unitary equivalence of the extended and reduced phase space quantizations of a constrained classical system with symmetry. Several examples are presented.

Geometric and Algebraic Reduction for Singular Momentum Maps

This paper concerns the reduction of singular constraint sets of symplectic manifolds. It develops a “geometric” reduction procedure, as well as continues the work of Sniatycki and Patrick on reduction a la Dirac. The relationships among the Dirac, geometric, and Sniatycki-Weinstein algebraic reduction procedures are studied. Primary emphasis is placed on the case where the constraints are given by the vanishing of a (singular) momentum map associated to the Hamiltonian action of a compact Lie group. Specifically, an explicit local normal form is given for the momentum map which is used to show that these various reductions are all well defined, and a necessary and sufficient condition for them to agree is derived. Related conditions are investigated and shown to be sufficient in the case of torus actions. Numerous examples are computed, illustrating the results. Some discussion and examples are given for the noncompact case.

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

Abstract In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradskiǐs theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.

A Multisymplectic Approach to the KdV Equation

A canonical multisymplectic formulation of the KdV equation is presented which, when space + time decomposed, gives rise to the usual symplectic description of dynamics on the appropriate space of Cauchy data. In addition to allowing one to treat the KdV equation covariantly, this formalism enables one to derive the Gardner symplectic structure for the KdV equation in a completely systematic way.

A class of non-polarizable symplectic manifolds

A class of compact 4-dimensional symplectic manifolds which admit no polarizations whatsoever is presented. These spaces also provide examples of nonparallelizable manifolds which are symplectic but have no complex, and hence no Kähler, structures.

Reduction of homogeneous Yang-Mills fields

Abstract The structure of the reduced phase space for a homogeneous Yang-Mills field on a spatially compactified ( n + 1 ) -dimensional Minkowski spacetime is studied. Using the theory developed in [AGJ], various reductions of this system are considered and are shown to agree. Moreover, the reduced phase space is realized as semi-algebraic set which carries a nondegenerate Poisson algebra. For the gauge groups SU( 2 ) or SO( 3 ) it is shown that this system is equivalent to that of n interacting particles moving in R 3 with zero total angular momentum. The particular cases n = 1 and 2 are discussed in detail.

Zero levels of momentum mappings for cotangent actions

Abstract We show that zero level sets of momentum mappings for cotangent actions are coisotropic, even when the momentum mapping is singular. The proof applies to both the finite- and infinite-dimensional cases. We use this result to show that vacuum Yang-Mills theory is indeed first class in the sense of Dirac.