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 uðx; tÞ and vðx; tÞ given by the following partial differential equations [21]:

Singular Lagrangians and Its Corresponding Hamiltonian Structures http://dx.doi.org/10.5772/66146 7

$$
\mu\_t + \mu u\_x + u\_{xxx} + \lambda \upsilon v\_x = 0 \tag{6}
$$

$$
\omega \upsilon\_t + \mu\_x \upsilon + \upsilon\_x \mu + \upsilon\_{xxx} = 0 \tag{7}
$$

where λ is a real parameter.

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-

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

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

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,

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

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

uðx; tÞ and vðx; tÞ given by the following partial differential equations [21]:

−1

<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

{φN; G} (5)

{F; G}DB ¼ −{F; φM}{φM; φN}

allow the construction of gauge systems which are completely integrable.

terms of the original Poisson bracket {, } on the full phase space by:

second class constraints, is always of full rank.

3. A parametric coupled KdV system

those difficulties will not be present.

evolution of observables.

localities in the final formulation.

6 Lagrangian Mechanics

where {φM; φN}

When discussing conserved quantities, we will assume that u and v belong to the real Schwartz space defined by

$$\mathcal{C}^{\circ}\_{\downarrow} = \left\{ w \in \mathbb{C}^{\circ}(\mathbb{R}) / \lim\_{x \to \pm \circ \circ} x^{p} \frac{\partial^{q}}{\partial x^{q}} w = 0; p, q \ge 0 \right\} \tag{8}$$

When λ ¼ þ1 the system is equivalent to two decoupled KdV equations. When λ ¼ −1 the system is equivalent to a KdV equation valued on the complex algebra. By a redefinition of v given by <sup>v</sup> ! <sup>v</sup>ffiffiffiffi <sup>j</sup>λ<sup>j</sup> <sup>p</sup> the system for <sup>λ</sup> <sup>&</sup>gt; 0 reduces to the <sup>λ</sup> ¼ þ1 case and the system for <sup>λ</sup> <sup>&</sup>lt; <sup>0</sup> reduces to the λ ¼ −1 case. The case λ ¼ 0 is an independent integrable system.

The system (6) and (7) for λ ¼ −1 describes a two-layer liquid model studied in references [17–19]. It is a very interesting evolution system. It is known to have solutions developing singularities on a finite time [24]. Also, a class of solitonic solutions was reported in [25] through the Hirota approach [26] and in [27] via a Bäcklund transformation in the sense of Wahlquist and Estabrook (WE) [28].

The system (6) and (7) for λ ¼ 0 correspond to the ninth Hirota-Satsuma [6] coupled KdV system given in Ref. [29] (for the particular value of k ¼ 0) (see also [30]) and is also included in the interesting study that relates integrable hierarchies with polynomial Lie algebras [31].

#### 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 w and y given by

$$\begin{aligned} u(\mathbf{x},t) &= w\_{\mathbf{x}}(\mathbf{x},t) \\ v(\mathbf{x},t) &= y\_{\mathbf{x}}(\mathbf{x},t) \end{aligned} \tag{9}$$

The system (6) and (7) rewritten in terms of w and y is given by

$$\begin{array}{ll} \mathfrak{w}\_{xt} + F[\mathfrak{w}, \mathfrak{y}] = 0, & F[\mathfrak{w}, \mathfrak{y}] = \mathfrak{w}\_{\mathfrak{x}}\mathfrak{w}\_{\mathfrak{xx}} + \mathfrak{w}\_{\mathfrak{x}\text{xxx}} + \lambda \mathfrak{y}\_{\mathfrak{x}}\mathfrak{y}\_{\mathfrak{x}\mathfrak{x}}\\ \mathfrak{y}\_{xt} + \mathsf{G}[\mathfrak{w}, \mathfrak{y}] = 0, & \mathsf{G}[\mathfrak{w}, \mathfrak{y}] = \mathfrak{w}\_{\mathfrak{x}\mathfrak{x}}\mathfrak{y}\_{\mathfrak{x}} + \mathsf{y}\_{\mathfrak{x}\mathfrak{x}}\mathfrak{w}\_{\mathfrak{x}} + \mathsf{y}\_{\mathfrak{x}\mathfrak{x}\mathfrak{x}}. \end{array} \tag{10}$$

We notice that the matrix constructed from the Frechet derivatives of F and G, with respect to w and y, is self-adjoint. We then conclude from the Helmholtz procedure that

$$\mathcal{L}\_1 = -\frac{1}{2} w\_x w\_t - \frac{1}{2} \lambda y\_x y\_t + \int\_0^1 (w F[\mu w, \mu y] + y \lambda G[\mu w, \mu y]) d\mu,\tag{11}$$

where λ ≠ 0, and

$$\mathcal{L}\_2 = -\frac{1}{2} w\_x y\_t - \frac{1}{2} w\_l y\_x + \int\_0^1 (yF[\mu w, \mu y] + wG[\mu w, \mu y]) d\mu,\tag{12}$$

for every real value of λ, are two Lagrangian densities which give rise, from a variational principle to Eqs. (6) and (7).

The Lagrangians associated with <sup>L</sup>i; <sup>i</sup> <sup>¼</sup> <sup>1</sup>; 2 are given by Liðw; <sup>y</sup>Þ ¼ <sup>ð</sup><sup>T</sup> 0 dtðþ<sup>∞</sup> −∞ dxLi; i ¼ 1; 2:

Independent variations of Li, for each i, with respect to w and y give rise to the field equations

$$\begin{cases} \delta\_w L\_i = 0\\ \delta\_y L\_i = 0 \end{cases} \tag{13}$$

which coincide, for each i, with Eqs. (6) and (7). In the above equations δ<sup>w</sup> and δ<sup>y</sup> denote the Gateaux functional variation defined by

$$\begin{split} \delta\_w L(w, y) &= \lim\_{\epsilon \to 0} \frac{L(w + \epsilon \delta w, y) - L(w, y)}{\epsilon} \\ \delta\_y L(w, y) &= \lim\_{\epsilon \to 0} \frac{L(w, y + \epsilon \delta y) - L(w, y)}{\epsilon} . \end{split} \tag{14}$$

The explicit expressions for L<sup>1</sup> and L<sup>2</sup> are given by

$$\mathcal{L}\_1 = -\frac{1}{2} w\_{\text{x}} w\_{\text{f}} - \frac{1}{6} w\_{\text{x}}{}^3 + \frac{1}{2} w\_{\text{xx}}{}^2 - \frac{\lambda}{2} w\_{\text{x}} y\_{\text{x}}{}^2 - \frac{\lambda}{2} y\_{\text{x}} y\_{\text{f}} + \frac{\lambda}{2} y\_{\text{xx}}{}^2,\tag{15}$$

$$\mathcal{L}\_2 = -\frac{1}{2} w\_x y\_t - \frac{1}{2} w\_t y\_x - \frac{1}{2} w\_x^2 y\_x - y\_x w\_{\text{xxx}} - \frac{\lambda}{6} y\_x^3. \tag{16}$$

The Lagrangians Li; i ¼ 1; 2, are singular Lagrangians, we thus expect a constrained Hamiltonian formulation associated with them. The same happens for the corresponding KdV Lagrangian that can be obtained from L<sup>1</sup> by imposing λ ¼ 0.

We consider first the Lagrangian L1. The conjugate momenta associated with w and y, which we denote by p and q, respectively, are given by

$$p = \frac{\partial \mathcal{L}\_1}{\partial w\_t} = -\frac{1}{2} w\_x, \quad q = \frac{\partial \mathcal{L}\_1}{\partial y\_t} = -\frac{\lambda}{2} y\_x. \tag{17}$$

We define

$$
\phi\_1 = p + \frac{1}{2} w\_x, \quad \phi\_2 = q + \frac{\lambda}{2} y\_x. \tag{18}
$$

Hence, φ<sup>1</sup> ¼ φ<sup>2</sup> ¼ 0 are constraints on the phase space. We then follow the Dirac procedure to determine the whole set of constraints. It turns out that these are the only constraints on the phase space.

The Hamiltonian density may be obtained directly from L<sup>1</sup> by performing a Legendre transformation,

$$\mathcal{H}\_1 = pw\_t + qy\_t \mathcal{L}\_1. \tag{19}$$

The Hamiltonian density is then given by

$$\mathcal{H}\_1 = \frac{1}{6}\boldsymbol{w}\_x^3 - \frac{1}{2}\boldsymbol{w}\_{xx}^2 + \frac{\lambda}{2}\boldsymbol{w}\_x\boldsymbol{y}\_x^2 - \frac{\lambda}{2}\boldsymbol{y}\_{xx}^2\tag{20}$$

and the Hamiltonian by H<sup>1</sup> ¼ ðþ<sup>∞</sup> −∞ dx H1:

We introduce a Poisson structure on the phase space by defining

$$\begin{cases} \{w(\mathbf{x}), p(\hat{\mathbf{x}})\}\_{PB} = \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{y(\mathbf{x}), q(\hat{\mathbf{x}})\}\_{PB} = \delta(\mathbf{x} - \hat{\mathbf{x}}) \end{cases} \tag{21}$$

with all other brackets between these variables being zero.

From them we obtain

L<sup>1</sup> ¼ − 1 2 wxwt− 1 2

L<sup>2</sup> ¼ − 1 2 wxyt − 1 2 wtyx þ

Gateaux functional variation defined by

The explicit expressions for L<sup>1</sup> and L<sup>2</sup> are given by

L<sup>2</sup> ¼ − 1 2 wxyt − 1 2 wtyx− 1 2 w2

Lagrangian that can be obtained from L<sup>1</sup> by imposing λ ¼ 0.

<sup>p</sup> <sup>¼</sup> <sup>∂</sup>L<sup>1</sup> ∂wt ¼ − 1 2

φ<sup>1</sup> ≡ p þ

1 2

L<sup>1</sup> ¼ − 1 2 wxwt− 1 6 wx 3 þ 1 2 wxx<sup>2</sup> − λ <sup>2</sup> wxyx 2 − λ <sup>2</sup> yxyt <sup>þ</sup>

we denote by p and q, respectively, are given by

where λ ≠ 0, and

8 Lagrangian Mechanics

equations

We define

principle to Eqs. (6) and (7).

λyxyt þ

The Lagrangians associated with Li; i ¼ 1; 2 are given by Liðw; yÞ ¼

δwLðw; yÞ ¼ lim

δyLðw; yÞ ¼ lim

e!0

e!0

ð1 0

ð1 0

for every real value of λ, are two Lagrangian densities which give rise, from a variational

Independent variations of Li, for each i, with respect to w and y give rise to the field

δwLi ¼ 0

which coincide, for each i, with Eqs. (6) and (7). In the above equations δ<sup>w</sup> and δ<sup>y</sup> denote the

The Lagrangians Li; i ¼ 1; 2, are singular Lagrangians, we thus expect a constrained Hamiltonian formulation associated with them. The same happens for the corresponding KdV

We consider first the Lagrangian L1. The conjugate momenta associated with w and y, which

wx; <sup>q</sup> <sup>¼</sup> <sup>∂</sup>L<sup>1</sup>

wx; φ<sup>2</sup> ¼ q þ

∂yt ¼ − λ

λ

Lðw þ eδw; yÞ−Lðw; yÞ e

Lðw; y þ eδyÞ−Lðw; yÞ

<sup>e</sup> :

<sup>x</sup>yx−yxwxxx−

ðwF½μw; μy� þ yλG½μw; μy�Þdμ, (11)

ðyF½μw; μy� þ wG½μw; μy�Þdμ, (12)

ðT 0 dt ðþ<sup>∞</sup> −∞

<sup>δ</sup>yLi <sup>¼</sup> <sup>0</sup> (13)

λ <sup>2</sup> yxx 2

λ 6 y3 dxLi; i ¼ 1; 2:

; (15)

<sup>x</sup>: (16)

<sup>2</sup> yx: (17)

<sup>2</sup> yx: (18)

(14)

$$\{\partial\_x^{\boldsymbol{\mu}} w(\mathbf{x}), \partial\_{\hat{\mathbf{x}}}^{\boldsymbol{\mu}} p(\hat{\boldsymbol{x}})\} = \partial\_{\mathbf{x}}^{\boldsymbol{\mu}} \partial\_{\hat{\mathbf{x}}}^{\boldsymbol{\mu}} \{w(\mathbf{x}), p(\hat{\boldsymbol{x}})\}. \tag{22}$$

It turns out that φ1; φ<sup>2</sup> are second class constraints. In fact,

$$\begin{cases} \{\phi\_1(\mathbf{x}), \phi\_1(\hat{\mathbf{x}})\}\_{PB} = \delta\_\mathbf{x}(\mathbf{x} - \hat{\mathbf{x}})\\ \{\phi\_1(\mathbf{x}), \phi\_2(\hat{\mathbf{x}})\}\_{PB} = 0\\ \{\phi\_2(\mathbf{x}), \phi\_2(\hat{\mathbf{x}})\}\_{PB} = \lambda \delta\_\mathbf{x}(\mathbf{x} - \hat{\mathbf{x}}). \end{cases} \tag{23}$$

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 F and G on phase space is defined by

$$\{F, G\}\_{\mathcal{DB}} = \{F, G\}\_{\mathcal{PB}} - \langle \{F, \phi\_i(\mathbf{x}')\}\_{\mathcal{PB}} \mathbb{C}\_{\vec{\eta}}(\mathbf{x}', \mathbf{x}') \{\phi\_j(\mathbf{x}'), G\}\_{\mathcal{PB}}\rangle\_{\mathbf{x}'} \}\_{\mathbf{x}'} \tag{24}$$

where <>x′ denotes integration on <sup>x</sup>′ from <sup>−</sup><sup>∞</sup> to <sup>þ</sup>∞. The indices <sup>i</sup>; <sup>j</sup> <sup>¼</sup> <sup>1</sup>; 2 and the <sup>C</sup>ijðx′ ; x″ Þ are the components of the inverse of the matrix whose components are fφ<sup>i</sup> ðx′ Þ,φ<sup>j</sup> ðx″ g PB.

This matrix becomes

$$
\begin{bmatrix}
\partial\_{\mathbf{x}'} \delta(\mathbf{x}' - \mathbf{x}'') & \mathbf{0} \\
\mathbf{0} & \lambda \partial\_{\mathbf{x}'} \delta(\mathbf{x}' - \mathbf{x}'')
\end{bmatrix}
\tag{25}
$$

and its inverse is given by

$$\left[\mathbb{C}\_{\vec{\eta}}(\mathbf{x}^{\prime},\mathbf{x}^{\prime})\right] = \begin{bmatrix} \int^{\mathbf{x}^{\prime}} \delta(\mathbf{s} - \mathbf{x}^{\prime}) d\mathbf{s} & \mathbf{0} \\\\ \mathbf{0} & \frac{1}{\lambda} \int^{\mathbf{x}^{\prime}} \delta(\mathbf{s} - \mathbf{x}^{\prime}) d\mathbf{s} \end{bmatrix}. \tag{26}$$

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

$$\begin{aligned} \{u(\mathbf{x}), u(\hat{\mathbf{x}})\}\_{D\mathbf{B}} &= -\eth\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}), \quad \{v(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{D\mathbf{B}} = -\frac{1}{\lambda} \eth\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{u(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{D\mathbf{B}} &= 0. \end{aligned} \tag{27}$$

We remind that this Poisson structure has been constructed assuming λ ≠ 0.

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

$$\begin{array}{l}\mu\_{t} = \{\mu, H\_{1}\}\_{DB} = -\mu\mu\_{x} - \mu\_{xxx} - \lambda\nu\upsilon\_{x} \\ \upsilon\_{t} = \{\upsilon, H\_{1}\}\_{DB} = -\mu\_{x}\upsilon - \upsilon\_{x}\mu - \upsilon\_{xxx}.\end{array} \tag{28}$$

We notice that adding any function of the constraints to H<sup>1</sup> does not change the result, since the Dirac bracket of the constraints with any other local function of the phase space variables is zero.

Using the above bracket relations for u and v, we may obtain directly the Dirac bracket of any two functionals Fðu; vÞ and Gðu; vÞ. We notice that the observables F and G may be functionals of w; y; p, and q, not only of u and v. In this sense, the phase space approach for singular Lagrangians provides the most general space of observables.

We now consider the action L<sup>2</sup> and its associated Hamiltonian structure. In this case, we denote the conjugate momenta to w and y by p^ and q ^, respectively. We have

$$
\hat{p} = -\frac{1}{2} y\_x, \quad \hat{q} = -\frac{1}{2} w\_x. \tag{29}
$$

In this case, the constraints become

$$
\widehat{\phi\_1} = \hat{p} + \frac{1}{2}y\_x = 0, \quad \widehat{\phi\_2} = \hat{q} + \frac{1}{2}w\_x = 0. \tag{30}
$$

The corresponding Poisson brackets are given by

$$\begin{aligned} \{\widehat{\phi\_1}(\mathbf{x}), \widehat{\phi\_1}(\mathbf{x'})\}\_{PB} &= 0, & \{\widehat{\phi\_2}(\mathbf{x}), \widehat{\phi\_2}(\mathbf{x'})\}\_{PB} &= 0, \\ \{\widehat{\phi\_1}(\mathbf{x}), \widehat{\phi\_2}(\mathbf{x'})\}\_{PB} &= \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \mathbf{x'}). \end{aligned} \tag{31}$$

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

Singular Lagrangians and Its Corresponding Hamiltonian Structures http://dx.doi.org/10.5772/66146 11

$$\begin{cases} \{\boldsymbol{\mu}(\mathbf{x}), \boldsymbol{\mu}(\hat{\mathbf{x}})\}\_{DB} = 0, & \{\boldsymbol{\nu}(\mathbf{x}), \boldsymbol{\nu}(\hat{\mathbf{x}})\}\_{DB} = 0, \\ \{\boldsymbol{\mu}(\mathbf{x}), \boldsymbol{\nu}(\hat{\mathbf{x}})\}\_{DB} = -\partial\_{\mathbf{x}}\delta(\mathbf{x} - \hat{\mathbf{x}}). \end{cases} \tag{32}$$

The Hamiltonian H<sup>2</sup> ¼ ðþ<sup>∞</sup> −∞ dx H<sup>2</sup> is given in terms of the Hamiltonian density

and its inverse is given by

10 Lagrangian Mechanics

<sup>½</sup>Cijðx′ ; x″ Þ� ¼

fuðxÞ,vðx^ÞgDB ¼ 0:

Lagrangians provides the most general space of observables.

denote the conjugate momenta to w and y by p^ and q

φ c<sup>1</sup> ¼ p^ þ

The corresponding Poisson brackets are given by

fφ c<sup>1</sup> ðxÞ,φ <sup>c</sup><sup>1</sup> <sup>ð</sup>x′

fφ c<sup>1</sup> ðxÞ,φ <sup>c</sup><sup>2</sup> <sup>ð</sup>x′

Poisson structure for the original variables

In this case, the constraints become

ðx′

<sup>δ</sup>ðs−x″

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

<sup>f</sup>uðxÞ,uðx^ÞgDB <sup>¼</sup> −∂xδðx−x^Þ; <sup>f</sup>vðxÞ,vðx^ÞgDB <sup>¼</sup> <sup>−</sup> <sup>1</sup>

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

ut ¼ fu; H1gDB ¼ −uux−uxxx−λvvx

We notice that adding any function of the constraints to H<sup>1</sup> does not change the result, since the Dirac bracket of the constraints with any other local function of the phase space variables is zero. Using the above bracket relations for u and v, we may obtain directly the Dirac bracket of any two functionals Fðu; vÞ and Gðu; vÞ. We notice that the observables F and G may be functionals of w; y; p, and q, not only of u and v. In this sense, the phase space approach for singular

We now consider the action L<sup>2</sup> and its associated Hamiltonian structure. In this case, we

^ ¼ − 1 2

c<sup>2</sup> ¼ q ^ þ 1 2

> c<sup>2</sup> ðxÞ,φ <sup>c</sup><sup>2</sup> <sup>ð</sup>x′

> > Þ:

We remind that this Poisson structure has been constructed assuming λ ≠ 0.

p^ ¼ − 1 2 yx, q

> 1 2

yx ¼ 0, φ

Þg PB <sup>¼</sup> <sup>0</sup>, <sup>f</sup><sup>φ</sup>

ÞgPB <sup>¼</sup> <sup>∂</sup>xδðx−x′

From them, we can construct the Dirac brackets after which some calculations yield the

Þds 0

λ ðx′

<sup>δ</sup>ðs−x″ Þds

vt ¼ fv; <sup>H</sup>1gDB <sup>¼</sup> <sup>−</sup>uxv−vxu−vxxx: (28)

^, respectively. We have

ÞgPB ¼ 0,

wx: (29)

wx ¼ 0: (30)

<sup>λ</sup> <sup>∂</sup>xδðx−x^<sup>Þ</sup>

5: (26)

(27)

(31)

<sup>0</sup> <sup>1</sup>

$$\mathcal{H}\_2 = \frac{1}{2} w\_x^2 y\_x + y\_x w\_{xxx} + \frac{\lambda}{6} y\_x^3. \tag{33}$$

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

$$\mu\_t = \{\mu, H\_2\}\_{\text{DB}}, \quad \upsilon\_t = \{\upsilon, H\_2\}\_{\text{DB}}, \tag{34}$$

which coincide with the field Eqs. (6) and (7) for any value of λ. We have thus constructed two Hamiltonian functionals and associated Poisson bracket structures. These two Hamiltonian structures arise directly from the basic actions L<sup>1</sup> and L2. In Section 6, we will construct two additional Hamiltonian structures by considering a Miura transformation for the coupled system.

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

We have then constructed two Lagrangian densities Li; i ¼ 1; 2; we may now introduce a real parameter k and define a parametric Lagrangian density

$$
\mathcal{L}\_k = k \mathcal{L}\_1 + (1 - k)\mathcal{L}\_2. \tag{35}
$$

The field equations obtained from the corresponding Lagrangian Lk ¼ ðT 0 dtðþ<sup>∞</sup> −∞ dxL<sup>k</sup> are equivalent to Eqs. (6) and (7) in the following cases: If λ < 0 for any k. If λ ¼ 0; for k ≠ 1: If λ > 0 for k ≠ <sup>1</sup> <sup>1</sup><sup>þ</sup> ffiffi λ <sup>p</sup> and k ≠ <sup>1</sup> <sup>1</sup><sup>−</sup> ffiffi λ <sup>p</sup> . From now on, we will exclude these particular values of k.

The parametric Lagrangian Lk is singular for any value of k (excluding the above mentioned particular cases). The corresponding phase space formulation contains constraints, which are determined by the use of the Dirac procedure. We denote p and q the conjugate momenta associated with w and y, respectively. From their definition, we obtain the primary constraints.

$$
\phi\_1 \equiv \frac{k}{2} w\_x + \frac{(1-k)}{2} y\_x + p = 0 \tag{36}
$$

$$
\phi\_2 \equiv \frac{\lambda k}{2} y\_x + \frac{(1-k)}{2} w\_x + q = 0. \tag{37}
$$

We may then define the Hamiltonian density H<sup>k</sup> through the Legendre transformation, we get

$$
\mathcal{H}\_k = pw\_t + qy\_t \\
$$

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

$$\begin{aligned} \{\phi\_1(\mathbf{x}), \phi\_1(\hat{\mathbf{x}})\}\_{PB} &= k \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{\phi\_2(\mathbf{x}), \phi\_2(\hat{\mathbf{x}})\}\_{PB} &= \lambda k \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{\phi\_1(\mathbf{x}), \phi\_2(\hat{\mathbf{x}})\}\_{PB} &= (1 - k) \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}). \end{aligned} \tag{39}$$

Hence, they are second class constraints. We will denote by fg<sup>k</sup> DB the Dirac bracket corresponding to the parameter k. We then proceed to calculate the Dirac brackets of the original fields u and v.

We obtain

$$\begin{aligned} \{u(\mathbf{x}), u(\hat{\mathbf{x}})\}\_{DB}^{k} &= \frac{\lambda k}{-\lambda k^{2} + (1 - k)^{2}} \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{v(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{DB}^{k} &= \frac{k}{-\lambda k^{2} + (1 - k)^{2}} \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{u(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{DB}^{k} &= \frac{1 - k}{-\lambda k^{2} + (1 - k)^{2}} \left(-\mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\right). \end{aligned} \tag{40}$$

where the denominator is different from zero for the values of k we are considering. The above brackets define the Poisson structure of the corresponding Hamiltonian

$$H\_k = \int\_{-\infty}^{+\infty} d\mathcal{X} \mathcal{H}\_k. \tag{41}$$

The Hamilton equations

$$\begin{aligned} \mu\_t &= \{\mu, H\_k\}\_{DB} \\ \upsilon\_t &= \{\upsilon, H\_k\}\_{DB} \end{aligned} \tag{42}$$

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 F and G that

$$\{F, G\}\_{\rm DB}^k = \frac{-\lambda k}{-\lambda k^2 + (1 - k)^2} \{F, G\}\_{\rm DB}^1 + \frac{1 - k}{-\lambda k^2 + (1 - k)^2} \{F, G\}\_{\rm DB}^0,\tag{43}$$

where <sup>f</sup>F; <sup>G</sup>g<sup>1</sup> DB, corresponding to <sup>k</sup> <sup>¼</sup> 1, and <sup>f</sup>F; <sup>G</sup>g<sup>0</sup> DB, corresponding to k ¼ 0, are the two Dirac brackets structures obtained in Section 3. In particular, for any <sup>λ</sup>≠0; 1 and <sup>k</sup> <sup>¼</sup> <sup>1</sup> <sup>1</sup>−λ, we get

$$\{F, G\}\_{DB}^k = \{F, G\}\_{DB}^1 + \{F, G\}\_{DB}^0,\tag{44}$$

which implies that any linear combination of <sup>f</sup>F; <sup>G</sup>g<sup>1</sup> DB and <sup>f</sup>F; <sup>G</sup>g<sup>0</sup> DB, for any λ≠0; 1, is a Poisson bracket. That is, the two Poisson structures obtained in Ref. [22], corresponding to k ¼ 1 and k ¼ 0, are compatible.

For the particular value of λ ¼ 0, and any k≠1 we obtain

H<sup>k</sup> ¼ pwt þ qyt

Hence, they are second class constraints. We will denote by fg<sup>k</sup>

fuðxÞ,uðx^Þg<sup>k</sup>

fvðxÞ,vðx^Þg<sup>k</sup>

<sup>f</sup>uðxÞ, <sup>v</sup>ðx^Þg<sup>k</sup>

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

DB <sup>¼</sup> <sup>−</sup>λ<sup>k</sup> −λk

<sup>f</sup>F; <sup>G</sup>g<sup>k</sup>

brackets define the Poisson structure of the corresponding Hamiltonian

show they are compatible. It follows, for any two functionals F and G that

<sup>2</sup> þ ð1−k<sup>Þ</sup>

DB, corresponding to <sup>k</sup> <sup>¼</sup> 1, and <sup>f</sup>F; <sup>G</sup>g<sup>0</sup>

that these are the only constraints in the formulation.

conjugate pairs are

12 Lagrangian Mechanics

original fields u and v.

The Hamilton equations

where <sup>f</sup>F; <sup>G</sup>g<sup>1</sup>

We obtain

We now follow the Dirac algorithm to determine the complete set of constraints. It turns out

The Poisson brackets of the constraints obtained from the canonical Poisson brackets of the

fφ1ðxÞ,φ1ðx^ÞgPB ¼ k∂xδðx−x^Þ fφ2ðxÞ,φ2ðx^ÞgPB ¼ λk∂xδðx−x^Þ fφ1ðxÞ,φ2ðx^ÞgPB ¼ ð1−kÞ∂xδðx−x^Þ:

corresponding to the parameter k. We then proceed to calculate the Dirac brackets of the

DB <sup>¼</sup> <sup>λ</sup><sup>k</sup>

DB <sup>¼</sup> <sup>k</sup> −λk

DB <sup>¼</sup> <sup>1</sup>−<sup>k</sup> −λk

Hk ¼

where the denominator is different from zero for the values of k we are considering. The above

ðþ<sup>∞</sup> −∞

ut ¼ fu; HkgDB vt ¼ fv; HkgDB

In Section 3, we constructed two Poisson structures for the coupled system (6) and (7). We now

<sup>2</sup> <sup>f</sup>F; <sup>G</sup>g<sup>1</sup>

Dirac brackets structures obtained in Section 3. In particular, for any <sup>λ</sup>≠0; 1 and <sup>k</sup> <sup>¼</sup> <sup>1</sup>

DB þ

−λk

1−k

<sup>2</sup> þ ð1−k<sup>Þ</sup>

<sup>−</sup>λk<sup>2</sup> þ ð1−k<sup>Þ</sup>

<sup>2</sup> þ ð1−k<sup>Þ</sup>

<sup>2</sup> þ ð1−k<sup>Þ</sup>

2 �

<sup>2</sup> ∂xδðx−x^Þ

<sup>2</sup> ∂xδðx−x^Þ

−∂xδðx−x^Þ

� :

dxHk: (41)

<sup>2</sup> <sup>f</sup>F; <sup>G</sup>g<sup>0</sup>

DB, corresponding to k ¼ 0, are the two

−L<sup>k</sup> ¼ kH<sup>1</sup> þ ð1−kÞH2: (38)

(39)

(40)

(42)

DB; (43)

<sup>1</sup>−λ, we get

DB the Dirac bracket

$$\{F, G\}\_{\text{DB}}^k = \frac{k}{2(1-k)^2} \{F, G\}\_{\text{DB}}^\frac{1}{} + \frac{1-2k}{(1-k)^2} \{F, G\}\_{\text{DB}}^0. \tag{45}$$

For <sup>k</sup> <sup>¼</sup> <sup>2</sup> <sup>5</sup> the two coefficients on the right-hand member of Eq. (45) are equal. It implies that the Poisson structures for <sup>k</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> and k ¼ 0 are compatible.

We have thus constructed a pencil of Poisson structures, except for λ ¼ 1, for which the coupled system reduces to two decoupled KdV equations.
