ANNALS OF PHYSICS 169, 2947 (1986) Covariant Poisson Brackets for Classical Fields J. E. MARSDEN AND R. MONTGOMERY Department of Mathematics, Universiry of California, Berkeley, California 94720 P. J. MORRISON Deparfmem of Physics and Institute jbr Fusion Studies, University qf Texas, Ausrin, Texas 78712 AND W. B. THOMPSON Department of Physics, University of California. San Diego, La Jolla, California 92093 Received January 8, 1985 Poisson brackets that are spacetime covariant are presented for a variety of relativistic field theories. These theories include electromagnetism, general relativity, and general relativistic fluids and plasmas in Eulerian representation. The examples presented suggest the develop- ment of a general theory; the beginnings of such a theory are presented. Our covariant bracket formalism provides a general setting for. amongst other things, clarifying the transition from the covariant formalism to the dynamical 3 + 1 Hamiltonian formalism of Dirac and Arnowitt, Deser, and Misner. We illustrate the relevant procedures with electromagnetism. 4? 1986 Academic Press. Inc. 1. INTRODUCTION The purpose of this paper is to show how to write the equations of some specific general relativistic field theories in covariant Poisson bracket form. Our approach is to proceed from explicit examples to some speculations on the structure of the underlying mathematical theory. For each of the examples, the field equations will be shown to be equivalent to equations of the form where F is an arbitrary function of the fields and S is an action integral. The theories considered fall into two categories: A. Purefields, typified by gauge fields, where F and S in (1.1) are functions of the basic field variables 4” and their conjugate momenta ~5. 29 ooo3-4916/86 s7.50 Copyright 0 1986 by Academic Press. Inc. All rights of reproduction in any form reserved.