**Singular Lagrangians and Its Corresponding Hamiltonian Structures** Singular Lagrangians and Its Corresponding

Alvaro Restuccia and Adrián Sotomayor Alvaro Restuccia and Adrián Sotomayor

Hamiltonian Structures

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66146

#### Abstract

We present a general procedure to obtain the Lagrangian and associated Hamiltonian structure for integrable systems of the Helmholtz type. We present the analysis for coupled Korteweg-de Vries systems that are extensions of the Korteweg-de Vries equation. Starting with the system of partial differential equations it is possible to follow the Helmholtz approach to construct one or more Lagrangians whose stationary points coincide with the original system. All the Lagrangians are singular. Following the Dirac approach, we obtain all the constraints of the formulation and construct the Poisson bracket on the physical phase space via the Dirac bracket. We show compatibility of some of these Poisson structures. We obtain the Gardner ε-deformation of these systems and construct a master Lagrangian which describe the coupled systems in the weak ε-limit and its modified version in the strong ε-limit.

Keywords: integrable systems, conservation laws, partial differential equations, rings and algebras
