## 1. Introduction

The Lagrangian mechanics has a wide range of applications from classical mechanics to quantum field theory. There are two main reasons to introduce a Lagrangian in order to describe a physical model. Its stationary points, defined in terms of functional derivatives, provide the classical equations of motion or classical field equations governing the evolution of the physical system while the action functional constructed from the Lagrangian provides the path integral approach to quantum mechanics and quantum field theories. In this chapter, we analyze several aspects of singular Lagrangians, which are relevant in various areas of physics. They are essential in the description of the fundamental forces in nature and in the analysis of integrable systems. In this chapter, we consider recent applications of singular Lagrangians in the area of completely integrable systems.

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 physical points of view.

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 gravity theory.

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

## 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

If the Hessian matrix

The system presents then constraints on the phase space defined by the conjugate pairs

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

From the equation defining momenta one obtains, in the case of singular Lagrangian, a set of constraints

Also, by performing a Legendre transformation one gets a Hamiltonian

where

(ii) determine Lagrange multipliers, or

(iii) give new constraints.

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 non-localities 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

where

The difficulty in field theory occurs when the matrix

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.

## 3. A parametric coupled KdV system

A very interesting and well‐known integrable system is the Korteweg‐de Vries (KdV) equation. It arises from a variational principle of a singular Lagrangian. In what follows, we consider an extension of it. A coupled KdV system formulated in terms of two real differentiable functions

where

When discussing conserved quantities, we will assume that

When

The system (6) and (7) for

The system (6) and (7) for

## 4. The Lagrangian associated with the parametric coupled KdV system

In this section, we obtain the Lagrangian and associated Hamiltonian structure of the coupled KdV system. We present the main results in Ref. [22].

The Lagrangian construction requires the introduction of the Casimir potentials

The system (6) and (7) rewritten in terms of

We notice that the matrix constructed from the Frechet derivatives of

where

for every real value of

The Lagrangians associated with

Independent variations of

which coincide, for each

The explicit expressions for

The Lagrangians

We consider first the Lagrangian

We define

Hence,

The Hamiltonian density may be obtained directly from

The Hamiltonian density is then given by

and the Hamiltonian by

We introduce a Poisson structure on the phase space by defining

with all other brackets between these variables being zero.

From them we obtain

It turns out that

In order to define the Poisson structure on the constrained phase space, we need to use the Dirac brackets.

The Dirac bracket between two functionals

where

This matrix becomes

and its inverse is given by

It turns out, after some calculations, that the Dirac brackets of the original variables are

We remind that this Poisson structure has been constructed assuming

From them, we obtain the Hamilton equations, which of course are the same as Eqs. (6) and (7):

We notice that adding any function of the constraints to

Using the above bracket relations for

We now consider the action

In this case, the constraints become

The corresponding Poisson brackets are given by

From them, we can construct the Dirac brackets after which some calculations yield the Poisson structure for the original variables

The Hamiltonian

The Hamilton equations follow then in terms of the Dirac brackets, they are

which coincide with the field Eqs. (6) and (7) for any value of

## 5. A pencil of Poisson structures for the parametric coupled KdV system

We have then constructed two Lagrangian densities

The field equations obtained from the corresponding Lagrangian

The parametric Lagrangian

We may then define the Hamiltonian density

We now follow the Dirac algorithm to determine the complete set of constraints. It turns out that these are the only constraints in the formulation.

The Poisson brackets of the constraints obtained from the canonical Poisson brackets of the conjugate pairs are

Hence, they are second class constraints. We will denote by

We obtain

(40) |

where the denominator is different from zero for the values of

The Hamilton equations

coincide, as expected, with the coupled Eqs. (6) and (7).

In Section 3, we constructed two Poisson structures for the coupled system (6) and (7). We now show they are compatible. It follows, for any two functionals

where

which implies that any linear combination of

For the particular value of

For

We have thus constructed a pencil of Poisson structures, except for

## 6. The Miura transformation for the parametric coupled KdV system

It is well known that the KdV equation admits two Hamiltonian structures, one of them is a particular case of our previous construction. It is obtained by considering only the

and the modified KdV system (MKdVS)

It is interesting that from Eq. (47), following the Helmholtz procedure, which is also valid for the MKdVS system, we obtain two singular Lagrangians densities

and

Eq. (48) being valid only for

Each of them has a Poisson structure that follows from the Dirac approach. The Dirac brackets, for the original fields

(50) |

which is the Poisson structure associated with

(51) |

the Poisson structure associated with

The corresponding Hamiltonian densities

The Hamilton equations obtained from these Hamiltonian structures coincide, of course, with Eqs. (6) and (7).

From these two Poisson structures, we may construct a pencil of Poisson structures as we described in the previous section, see Ref. [22] for the details of the construction. We notice that

## 7. A duality relation among the Lagrangians of the parametric coupled KdV system

We consider a generalization of the Gardner construction for the KdV equation. The Gardner transformation for the system (6) and (7) is given by

where

Any solution of Eqs. (55) and (56) define through Eqs. (53) and (54) a solution of the system (6), (7).

If we consider the

and rewrite Eqs. (53) and (54), we get

Taking the limit

which is exactly the Miura transformation. In the same limit, we obtain from Eqs. (55), (56) the Miura equations given by Eq. (47).

We now construct using the Helmholtz approach a master Lagrangian for the Gardner equations. The master Lagrangians, there are two of them, are

We introduce the Casimir potentials

and using the Helmholtz approach we obtain the Lagrangian densities

(64) |

If we take the weak coupling limit

where

If we redefine

and take the strong coupling limit

where

## 8. Hamiltonian structure for a KdV system valued on a Clifford algebra

In this section, we continue the discussion of the Lagrangian and Hamiltonian structures for the coupled KdV systems. We discuss a coupled system arising from the breaking of the supersymmetry on the

where

and

where

We proposed in Ref. [15] the following coupled KdV system arising from the breaking of the supersymmetry in the

In distinction to the

(74) |

It is interesting to remark that the following nonlocal conserved charge of Super KdV [32] is also a nonlocal conserved charge for the system (73), in terms of the Clifford algebra valued field

However, the nonlocal conserved charges of Super KdV in Ref. [33] are not conserved by the system (73). For example,

is not conserved by Eq. (73).

The system (73) has multisolitonic solutions. In Ref. [34], we showed that the soliton solution is Liapunov stable under perturbation of the initial data.

## 9. The Lagrangian and Hamiltonian structure of the Clifford valued system

We introduce the Casimir potentials

We notice, as in the previous sections, that Eq. (73) may be expressed as stationary points of a singular Lagrangian constructed following the Helmholtz approach. We denote

The Lagrangian becomes

From the Lagrangian

Eq. (80) describes constraints on the phase space.

Performing the Legendre transformation we obtain the Hamiltonian of the system

where

Following the Dirac approach, the conservation of the primary constraints (80) determines the Lagrange multipliers associated with the constraints (80). There are no more constraints on the phase space. It turns out that both constraints are second class ones. The Poisson structure of the constrained Hamiltonian is then determined by the Dirac brackets, see Ref. [15] for the details. We identify by an index

Introducing

The Poisson structure of the constrained Hamiltonian is then determined by the Dirac brackets [20]. For any two functionals on the phase space

where

We then have

Consequently,

where

## 10. Positiveness of the Hamiltonian for the Clifford valued system

An interesting property of the Hamiltonian *a priori* positiveness. In fact,

where the Sobolev norm

We also noticed that

where

We then have

We now use the bound

to obtain

Consequently,

Finally,

Hence, for a normalized state satisfying

The Hamiltonian is then manifestly bounded from below in the space of normalized

The property of the Hamiltonian is relevant from the physical point of view. In particular, in showing that the soliton solution of the Clifford coupled system is Liapunov stable. The stability analysis follows ideas introduced in Ref. [35] for the KdV equation. It is based on the use of the conserved quantities of the system. It is interesting that only the first few of them, in the case of the KdV equation, are needed. In the case of the Clifford coupled system these are all the local conserved quantities of the system.

## 11. The KdV equation valued on the octonion algebra

A famous theorem by Hurwitz establishes that the only real normalized division algebras are the reals

The extension of the KdV equation to a partial differential equation for a field valued on a octonion algebra is then an interesting goal [23].

We showed in the previous sections that an extension of the KdV equation to the field valued on a Clifford algebra give rise to a coupled system with Liapunov stable soliton solution but without an infinite sequence of local conserved quantities.

In the present section, we analyze the KdV extension where the field is valued on the octonion algebra. The system shares several properties of the original real KdV equation. It has soliton solutions and also has an infinite sequence of local conserved quantities derived from a Bäcklund transformation and a bi‐Lagrangian and bi‐Hamiltonian structure [23]. We will show in this section the construction of the bi‐Lagrangian structure.

The octonion algebra contains as subalgebras all other division algebras, hence our construction may be reduced to any of them.

The KdV equation on the octonion algebra can be seen as a coupled KdV system, as we will see it has some similarities to the construction in the previous sections. However, it is invariant under the exceptional Lie group

We denote

where

The KdV equation formulated on the algebra of octonions, or simply the octonion KdV equation, is given by

when

Eq. (98) is invariant under the Galileo transformation given by

where

Additionally, Eq. (98) is invariant under the automorphisms of the octonions, that is, under the group

then

and consequently

## 12. The Gardner formulation for the octonion valued algebra KdV equation

Associated with the real KdV equation, there is a Gardner

The generalized Gardner equation is then

where

If

## 13. The master Lagrangian for the KdV equation valued on the octonion algebra

We may now use the Helmholtz procedure to obtain a Lagrangian density for the generalized Gardner equation. The master Lagrangian formulated in terms of the Casimir potential

is

where the Lagrangian density is given by

The Lagrangian density

Independent variations with respect to

(110) |

Using properties of the octonion algebra we obtain from the stationary requirement

In the calculation the property to be a division algebra of the octonions is explicitly used.

If we take the limit

Independent variations with respect to

and take the limit

where

We get in this limit the generalized Miura Lagrangian

The Miura equation is then obtained by taking variations with respect to

while the Miura transformation arises after the redefinition process, it is

Any solution of the Miura equation, through the Miura transformation, yields a solution of the KdV equation valued on the octonion algebra. Since

The Lagrangian formulation of the octonionic KdV equation may be used as the starting step to obtain the Hamiltonian structure of the octonion algebra valued KdV equation.

## 14. Conclusions

We analyzed the relevance of the Dirac approach for constraint systems applied to singular Lagrangians. Several interesting theories are described by singular Lagrangians, notoriously the gauge theories describing the known fundamental forces in nature. In this chapter, we emphasized its relevance in the formulation of completely integrable field theories. We discussed extensions of the Korteweg‐de Vries equation in different contexts. All these extensions, together with the KdV equation, allow a construction of a Lagrangian and a Hamiltonian structure arising from the application of the Helmholtz procedure. That is, starting with a time evolution partial differential system we construct, following the Helmholtz procedure, a Lagrangian associated with it. We present the construction of several Lagrangians and their corresponding Hamiltonian structures associated with the coupled KdV systems. All of them are characterized by second class constraints. The physical phase space is obtained by the determination of the complete set of constraints and the corresponding Dirac brackets. We established the relation between the several constructions by obtaining a pencil of Poisson structures. The application includes systems with an infinite sequence of conserved quantities together with a system with finite number of conserved quantities but presenting soliton solutions with nice stability properties. The final application is an extension of the KdV equation to the case in which the fields are valued on the octonion algebra. We constructed a master formulation from which two dual Lagrangian formulations are obtained, one corresponding to the KdV valued on the octonions and the other one corresponding to the extension of the modified KdV equation to fields valued on the octonions.

One important extrapolation of the analysis we have presented is the construction of gauge theories describing completely integrable systems. In fact, it is natural to extend the analysis by constructing a gauge theory which under a gauge fixing procedure reduces to the completely integrable systems of the KdV type we have discussed.

## 15. Acknowledgments

A. R. and A. S. are partially supported by Project Fondecyt 1161192, Chile.

**Pacs**: 02.30.lk, 11.30.-j, 02.30. Jr, 02.10.Hh