Dirac's theory of constraints in field theory and the canonical form of Hamiltonian differential operators

In the classical theory of Hamiltonian systems, great emphasis is placed on the introduction of canonical coordinates-the positions and conjugate momenta of classical mechanics. I Canonical coordinates serve to simplify many of the equations and transformations required in the study of finite-dimensional Hamiltonian systems. Most quantization procedures require that the Hamiltonian system be in canonical form before proceeding. Hamiltonian perturbation theories are much easier to develop in canonical coordinates.2• However, in recent years there has been a renewed interest in Hamiltonian systems in noncanonical coordinates. The principle motivation has been the development of an infinite-dimensional theory of Hamiltonian systems of evolution equations in which the role of the skew-symmetric symplectic matrix J is played by a skew-adjoint Hamiltonian differential operator, and the Hamiltonian function is replaced by a Hamiltonian functional. 4,5 Applications to stability questions in fluid mechanics and plasma physics and also to completely integrable (soliton) equations,8 have been just a few of the important consequences of this general theory. A significant open problem in this theory is the Darboux problem of whether one can always determine suitable canonical coordinates for such a Hamiltonian system. In this paper, a general result of this type for constant coefficient Hamiltonian differential operators is proved, along with some extensions of the result to more general field-dependent Hamiltonian operators. In the case of finite-dimensional Hamiltonian systems, Darboux' theorem guarantees that canonical coordinates can always be found, provided that the Poisson bracket has constant rank. For maximal rank (symplectic) Poisson brackets, the proof of Weinstein 10 is especially appealing in that it readily extends to certain infinite-dimensional situations. There are two main steps in Weinstein's proof: first the Hamiltonian operator is reduced to a constant operator by a clever change of variables; second, one shows that any constant-coefficient skew-adjoint operator can be placed into canonical form. In this light, the present paper can be viewed as an implementation of the second part of Weinstein's proof in the case of constant-coefficient skew-adjoint differential operators. The first part of the proof is far more difficult, and, unfortunately, the infinite-dimensional version of Darboux' theorem due to Weinstein does not appear to be applicable to the Hamiltonian differential operators of interest. The problem is that Weinstein requires some form of Banach manifold structure to effect his proof, but for differential operators that depend on the dependent variables it is not at all obvious how to impose such a structure. Even if one could mimic Weinstein's proof, the resulting changes of variable would be horribly nonlocal, and therefore be of limited use. Thus the question of whether Darboux' theorem is valid for Hamiltonian differential operators remains an important open problem. Only in special cases, including firstand third-order scalar operators, and some first-order matrix operators is the answer known. II ,26 (Results of Dubrovin and Novikov l2 indicate that Darboux' theorem may not hold for matrix operators involving more than one independent variable, but they only consider a limited class of changes of variable, so the general Darboux problem remains unanswered.) The underlying motivation of this paper can be found in the recent applications of Dirac's theory of constraints by Nutku to produce canonical forms of a number of Hamiltonian systems of evolution equations of physical interest, including the equations of shallow water waves and gas dynamics13 and the Korteweg-de Vries equation. 14 In the finite-dimensional theory of the calculus of variations, for nondegenerate Lagrangians the passage from the Euler-Lagrange equations to the corresponding canonical form of Hamilton's equations is classical. I Dirac's theory of constraints was designed to handle degenerate Lagrangians and produce canonical Hamiltonian systems, which, when subjected to the appropriate constraints, reduce to the original Euler-Lagrange equations. 15 In Nutku's applications of this theory, one begins with a Hamiltonian system of evolution equations, whose Poisson bracket is not in canonical form. The next step is to replace the original Hamiltonian system of evolution equations by an equivalent system of EulerLagrange equations; this appears to require that the Hamiltonian operator be constant coefficient. The resulting Lagrangian function is inevitably degenerate, so to construct a corresponding canonical Hamiltonian system one is required to invoke the Dirac machinery. The details of the construction can be found in Refs. 13 and 14. However, given the fact that one begins with a (noncanonical) Hamiltonian system, the entire procedure seems to be a bit roundabout, and it would be useful to have a direct method of constructing canonical Hamiltonian systems from more general Hamiltonian evolution equations. In this paper a simple constructive procedure for effecting this