2. The Dirac procedure for constrained systems

The Dirac approach for constrained systems [20] is a fundamental tool in the analysis of classical and quantum aspects of a physical theory. From a classical point of view, it provides a precise formulation of the initial valued problem for a time evolution system of partial differential equations. The initial data for the initial valued problem, given in terms of a constrained submanifold of a phase space, defines the physical phase space provided with the corresponding Poisson structure which gives rise to the canonical quantization of the system. In field theory, the starting point is a Lagrangian formulation. Its stationary points determine the classical field equations, generically a time evolution system of partial differential equations. From the Lagrangian density L, one defines the conjugate momenta pi ; i ¼ 1, …; N; associated with the original independent fields qi ; i ¼ 1, …; N; defining the Lagrangian:

$$p\_i = \frac{\partial \mathcal{L}}{\partial \dot{\eta}\_i}.\tag{1}$$

L is assumed to be a function of q\_ <sup>i</sup> and a finite number of spatial derivatives. If the Hessian matrix <sup>∂</sup>2<sup>L</sup> ∂q\_ i ∂q\_ j is singular we cannot express, from the above equation defining the conjugate momenta, all the q\_<sup>i</sup> velocities in terms of the conjugate momenta.

The system presents then constraints on the phase space defined by the conjugate pairs ðqi ; pi Þ, i ¼ 1, …; N: The phase space is provided with a Poisson structure given by

$$\{q\_i, \ p\_j\} = \delta\_{\vec{\eta}}, \quad \{q\_i, q\_j\} = \{p\_i, p\_j\} = 0. \tag{2}$$

In general, it is a difficult task to disentangle all the constraints on the phase space associated with a given Lagrangian. The Dirac approach provides a systematic way to obtain all the constraints on phase space. Moreover, it determines the Lagrange multipliers associated with the constraints (eventually after a gauge fixing procedure) in a way that if the constraints are satisfied initially then the Hamilton equations ensure that they are satisfied at any time. In this sense, it provides a precise formulation of the initial value problem, the initial data is given by the set of ðqi ; pi Þ conjugate pairs satisfying the constraints on phase space. The Hamilton equations then provide the time evolution of the system. This constrained initial data, with its associated Poisson structure (also obtained from the Dirac construction) provides the fundamental structure to define the canonical quantization of the original Lagrangian.

From the equation defining momenta one obtains, in the case of singular Lagrangian, a set of constraints φMðq; pÞ ¼ 0, where the argument is a shorthand notation for p; q and their derivatives with respect to the spatial coordinates xa; a ¼ 1, …; k:

Also, by performing a Legendre transformation one gets a Hamiltonian H<sup>0</sup> ¼ ðþ<sup>∞</sup> −∞ dxH0, where the Hamilton density is given by

$$\mathcal{H}\_0 = \sum\_i p\_i \dot{q}\_i \mathcal{L},\tag{3}$$

where L is the Lagrangian density. Then, we obtain a new Hamiltonian H ¼ ðþ<sup>∞</sup> −∞ dxH with a density H ¼ H<sup>0</sup> þ λMφM. The conservation of the constraints, which have to be satisfied at any time, yields

$$
\dot{\phi}\_M = \{\phi\_M, H\} = 0.\tag{4}
$$

{φM; H} ¼ 0 may (i) be identically satisfied on the constrained surface φ<sup>m</sup> ¼ 0,


The analysis of integrable systems, in particular the Korteweg-de Vries equation and extensions of it [1–16], have provided a lot of interesting results from both mathematical and

Besides the physical applications of coupled KdV systems at low energies [17–19], one of the Poisson structures of the KdV equation is related to the Virasoro algebra with central terms. The latest is a fundamental symmetry of string theory, a proposal for a consistent quantum

In this chapter, we discuss a general approach based on the Helmholtz procedure to obtain a Lagrangian formulation and the Hamiltonian structure, starting from the system of time evolution partial differential equations describing the coupled KdV systems. Once the Lagrangian, whose stationary points corresponds to the integrable equations, has been obtained we follow the Dirac approach to constrained systems [20] to obtain the complete set of constraints and the Hamiltonian structure of the system. We discuss the existence of more than one Poisson structures associated with the integrable systems. Some of them are compatible Poisson structures and define a pencil of Poisson structures. We also discuss duality relations among the integrable systems we consider. The extensions of the KdV equation include a parametric coupled KdV system [21, 22], which we discuss in Section 3. In Section 8, we present a coupled KdV system arising from the breaking of a N ¼ 1 supersymmetric model [15]. In Section 11, we discuss an extension of the KdV equation where the fields are valued on the octonion algebra and the product in the equation is the product on the octonion algebra [23]. This system has a supersymmetric extension which may be directly related to a model of the D ¼ 11 supermembrane theory, a relevant sector of M−theory. The latest is a proposal of

The Dirac approach for constrained systems [20] is a fundamental tool in the analysis of classical and quantum aspects of a physical theory. From a classical point of view, it provides a precise formulation of the initial valued problem for a time evolution system of partial differential equations. The initial data for the initial valued problem, given in terms of a constrained submanifold of a phase space, defines the physical phase space provided with the corresponding Poisson structure which gives rise to the canonical quantization of the system. In field theory, the starting point is a Lagrangian formulation. Its stationary points determine the classical field equations, generically a time evolution system of partial differential equations. From the Lagrangian density L, one defines the conjugate momenta

> pi <sup>¼</sup> <sup>∂</sup><sup>L</sup> ∂q\_i

<sup>i</sup> and a finite number of spatial derivatives.

; i ¼ 1, …; N; defining the

: (1)

unification of all fundamental forces at very high energy.

2. The Dirac procedure for constrained systems

; i ¼ 1, …; N; associated with the original independent fields qi

physical points of view.

gravity theory.

4 Lagrangian Mechanics

pi

Lagrangian:

L is assumed to be a function of q\_

In Case (i) or (ii), the procedure ends; in Case (iii), the iteration follows exactly in the same way. At some step, the procedure ends, assuming that there is a finite of physical degrees of freedom describing the dynamics of the original Lagrangian. In the procedure, a set of Lagrange multipliers may be determined and others may not. The constraints associated with the ones that have been determined are called second class constraints, the other constraints for which the Lagrange multipliers are not determined are related to first class constraints. The first class constraints are the generators of a gauge symmetry on the time evolution system of partial differential equations. A difficult situation may occur in field theory when there is a combination of first and second class constraints. In order to separate them, one may have to invert some matrix involving fields of the formulation which may render dangerous nonlocalities in the final formulation.

All physical theories of the known fundamental forces in nature are formulated in terms of Lagrangians with gauge symmetries. All of them have first class constraints in their canonical formulation. In addition, they may also have second class constraints. In the analysis of field theories which are completely integrable systems like the ones we will discuss in this chapter only second class constraint appear. In this case, there are short cut procedures to simplify the Dirac procedure. However, the richness of the Dirac approach is that from its formulation one can extrapolate gauge systems which under a gauge fixing procedure reduce to the given system with second class constraints only. This is one of the main motivations of this chapter, to establish the Lagrangian and Hamiltonian structure for coupled KdV systems, which may allow the construction of gauge systems which are completely integrable.

In the case in which the constrained system has second class constraints, Dirac introduced the Poisson structure on the constrained submanifold in phase space. It determines the "physical" phase space with its Poisson bracket structure given by the Dirac bracket. They are defined in terms of the original Poisson bracket {, } on the full phase space by:

$$\{F, G\}\_{DB} = -\{F, \phi\_M\} \{\phi\_M, \phi\_N\}^{-1} \{\phi\_N, G\} \tag{5}$$

where {φM; φN} <sup>−</sup><sup>1</sup> is the inverse of the matrix {φM; <sup>φ</sup>N} which, in the case where <sup>φ</sup><sup>M</sup> <sup>¼</sup> 0 are second class constraints, is always of full rank.

The difficulty in field theory occurs when the matrix {φM; φN} depends on the fields describing the theory and its inverse may lead to nonlocalities in the formulation. In our applications, those difficulties will not be present.

The Dirac bracket of a second class constraint with any other observable is zero. Consequently, the time conservation of the second class constraints is assured by the construction. For the same reason, there is no ambiguity on which Hamiltonian is used in determining the time evolution of observables.
