14. Conclusions

LεðsÞ ¼ Re −

Independent variations with respect to s yields

1 2 sxðδsÞ<sup>t</sup> − 1 6 �

LðwÞ ¼

tion (98). If we consider the following redefinition

and take the limit e ! ∞ we obtain

ðtf ti dt ðþ<sup>∞</sup> −∞

� þ 1 <sup>72</sup> <sup>ε</sup><sup>2</sup> �

1 2 ðδsÞxst−

ðδsÞxxsxx þ ðδsÞxxsxx

the generalized Gardner equation (106).

δLεðsÞ ¼ Re −

24 Lagrangian Mechanics

octonion algebra,

where

þ Re 1 2 � 1 2 sxst− 1 6 ðsxÞ 3 þ 1 2 ðsxxÞ 2 þ 1 <sup>72</sup> <sup>ε</sup><sup>2</sup> ðsxÞ 4

� �

<sup>2</sup> <sup>þ</sup> sxðδsÞxsx þ ðsx<sup>Þ</sup>

<sup>3</sup> <sup>þ</sup> sxðδsÞxðsx<sup>Þ</sup>

� �

2 ðδsÞ<sup>x</sup>

<sup>2</sup> þ ðsx<sup>Þ</sup> 2

<sup>L</sup>εðsÞ ! <sup>ε</sup><sup>2</sup>Lεð^s<sup>Þ</sup> (112)

<sup>L</sup>eð^sÞ ¼ <sup>L</sup><sup>M</sup>ð^sÞ, (113)

dxL<sup>M</sup>ð^sÞ: (115)

<sup>x</sup> ¼ 0; ^r ≡ ^sx, (116)

The Lagrangian density LεðsÞ is invariant under the action of the exceptional Lie group G2.

ðδsÞxðsxÞ

� � �

ðδsÞxðsxÞ

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

dxRe − 1 2 wxwt− 1 6 ðwxÞ 3 þ 1 2 ðwxxÞ 2

� � �

Using properties of the octonion algebra we obtain from the stationary requirement δLεðsÞ ¼ 0

If we take the limit e ! 0, we obtain a first Lagrangian for the KdV equation valued on the

Independent variations with respect to w yields, using u ¼ wx, the octonionic KdV equa-

s ! ^s ¼ εs

lim<sup>e</sup>!<sup>∞</sup> <sup>e</sup><sup>2</sup>

1 2 ^sx^st þ 1 2 ð^sxxÞ 2 þ 1 <sup>72</sup> <sup>ð</sup>^sx<sup>Þ</sup> 4

ðtf ti dt ðþ<sup>∞</sup> −∞

<sup>18</sup> <sup>ð</sup>^r<sup>Þ</sup> 3

LMð^sÞ ¼

^rt <sup>þ</sup> ^rxxx<sup>−</sup> <sup>1</sup>

The Miura equation is then obtained by taking variations with respect to ^s, we get

� �

<sup>L</sup><sup>M</sup>ð^sÞ ¼ <sup>R</sup><sup>e</sup> <sup>−</sup>

We get in this limit the generalized Miura Lagrangian

: (109)

3 ðδsÞ<sup>x</sup>

: (111)

: (114)

:

(110)

ðδsÞxsx þ ðsxÞ

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.
