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

We introduce the Casimir potentials w and η defined by

$$
\mu = w\_{\mathbf{x}} \text{ and } \xi = \eta\_{\mathbf{x}}.\tag{77}
$$

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

$$\begin{aligned} P(w,\eta) &= w\_{\text{xxxx}} + w\_{\text{x}}w\_{\text{xx}} + \frac{1}{4}(\mathcal{P}(\eta\_{\text{x}}\overline{\eta}\_{\text{x}}))\_{\text{x}} \\ Q(w,\eta) &= \eta\_{\text{xxx}} + \frac{1}{2}(w\_{\text{x}}\eta\_{\text{x}})\_{\text{x}} \end{aligned} \tag{78}$$

The Lagrangian becomes L ¼ ðT 0 dtðþ<sup>∞</sup> −∞ dxL in terms of the Lagrangian density L given by

$$\mathcal{L} = \frac{1}{2} w\_x w\_t + \frac{1}{2} (\mathcal{P}(\eta\_x \overline{\eta}\_t) - \int\_0^1 w \mathcal{P}(\mu w, \,\mu \eta) d\mu - \int\_0^1 \mathcal{P}(Q(\mu w, \,\mu \eta)\overline{\eta}) d\mu. \tag{79}$$

From the Lagrangian L, we may construct its Hamiltonian structure using the Legendre transformation. We denote ðp; σÞ the conjugate momenta to ðw; ηÞ:

$$\begin{aligned} p &:= \frac{\partial \mathcal{L}}{\partial(\partial\_t w)} = \frac{1}{2} w\_x = \frac{1}{2} u \\ \sigma &:= \frac{\partial \mathcal{L}}{\partial(\partial\_t \eta)} = \frac{1}{2} \eta\_x = \frac{1}{2} \varphi. \end{aligned} \tag{80}$$

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

Performing the Legendre transformation we obtain the Hamiltonian of the system

$$H = \int\_{-\infty}^{+\infty} d\mathbf{x} \left( p w\_t + \mathcal{P}(\sigma \overline{\eta}\_t) - \mathcal{L} \right) \tag{81}$$

where <sup>H</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>H</sup>^ <sup>5</sup> in (74).

In distinction to the N ¼ 1 supersymmetric KdV equation the coupled system (73) has only a

uPðξξÞþðuxÞ

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 ξ,

However, the nonlocal conserved charges of Super KdV in Ref. [33] are not conserved by the

The system (73) has multisolitonic solutions. In Ref. [34], we showed that the soliton solution is

We notice, as in the previous sections, that Eq. (73) may be expressed as stationary points of a

1 2 ðwxηxÞ<sup>x</sup>

1 4

ðPðηxηxÞÞ<sup>x</sup>

dxL in terms of the Lagrangian density L given by

singular Lagrangian constructed following the Helmholtz approach. We denote

Pðw; ηÞ ¼ wxxxx þ wxwxx þ

Qðw; ηÞ ¼ ηxxxx þ

ðT 0 dt ðþ<sup>∞</sup> −∞

9. The Lagrangian and Hamiltonian structure of the Clifford valued

� �

<sup>2</sup> <sup>þ</sup> <sup>P</sup>ðξxξx<sup>Þ</sup>

dx:

ξðsÞdsdx: (75)

ξðsÞdsdx: (76)

u ¼ wx and ξ ¼ ηx: (77)

(74)

(78)

finite number of local conserved quantities,

ðþ<sup>∞</sup> −∞ ξdx;

ðþ<sup>∞</sup> −∞ udx;

ðþ<sup>∞</sup> −∞ �

ðþ<sup>∞</sup> −∞ − 1 3 u3 − 1 2

Liapunov stable under perturbation of the initial data.

We introduce the Casimir potentials w and η defined by

<sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>P</sup>ðξξ<sup>Þ</sup>

ð∞ −∞ ξðxÞ ðx −∞

ð∞ −∞ uðxÞ ðx −∞

� dx;

H^ 1 <sup>2</sup> ¼

18 Lagrangian Mechanics

<sup>H</sup>^ <sup>1</sup> <sup>¼</sup>

<sup>V</sup>≡H^ <sup>3</sup> <sup>¼</sup>

<sup>M</sup>≡H^ <sup>5</sup> <sup>¼</sup>

system (73). For example,

is not conserved by Eq. (73).

The Lagrangian becomes L ¼

system

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 i the independent components of a field η or σ valued on the Clifford algebra. We may rewrite the constraints as

$$\begin{aligned} \upsilon &:= p - \frac{1}{2} w\_x \\ \upsilon\_i &:= \sigma\_i - \frac{1}{2} \eta\_{ix} . \end{aligned} \tag{82}$$

Introducing vI :¼ ðv; viÞ, we then have

$$\{\upsilon\_l(\mathbf{x}), \upsilon\_l(\mathbf{x'})\} = -\delta\_{l\dot{l}}\partial\_{\mathbf{x}}\delta(\mathbf{x} - \mathbf{x'}).\tag{83}$$

The Poisson structure of the constrained Hamiltonian is then determined by the Dirac brackets [20]. For any two functionals on the phase space F and G, the Dirac bracket is defined as

$$\{F, G\}\_{\mathcal{DB}} := \{F, G\} - \langle \{F, \upsilon\_l(\mathbf{x'})\} \{\upsilon\_l(\mathbf{x'}), \upsilon\_l(\mathbf{x'})\}^{-1} \rangle\_{\mathbf{x'}} \{\upsilon\_l(\mathbf{x'}), G\} \rangle\_{\mathbf{x'}},\tag{84}$$

where

$$\langle \{ v\_l(\mathbf{x'}), v\_l(\mathbf{x'}) \}^{-1} g(\mathbf{x'}) \rangle\_{\mathbf{x'}} = -\delta\_{l\parallel} \int\_{-\infty}^{\mathbf{x'}} g(\tilde{\mathbf{x}}) d\tilde{\mathbf{x}}.\tag{85}$$

We then have

$$\begin{aligned} \{\mu(\mathbf{x}), \mu(y)\}\_{DB} &= \eth\_{\mathbf{x}} \delta(\mathbf{x}, y), \\ \{\!\!\/\!\/\!\/\!\/\!\/\!\/, \phi\_{\not\!\/\!\/}(y)\}\_{DB} &= \eth\_{\not\!\/\!\/\!\/} \delta(\mathbf{x}, y), \\ \{\!\!\/\!\/\!\/\!\/\!\/)\_{DB} &= 0. \end{aligned} \tag{86}$$

Consequently,

$$\begin{aligned} \mathfrak{d}\_t u &= \{u, H\}\_{DB} = -\frac{1}{2} (u^2)\_x - u\_{\text{xxx}} - \frac{\lambda}{4} (\varphi\_i^2)\_x \\ \mathfrak{d}\_t \mathfrak{q}\_i &= \{\mathfrak{q}\_i, H\}\_{DB} = -\mathfrak{q}\_{i\text{xxx}} - \frac{\lambda}{2} (u \varphi\_i)\_x, \end{aligned} \tag{87}$$

where H is given by the last conserved quantity in Eq. (74) and can be directly expressed in terms of u and ξ.
