ABSTRACT
One of the most important properties of completely integrable non-linear partial differential equations is that they are Hamiltonian system. This chapter discusses how the bi-Hamiltonian structure can be deduced for a given non-linear system. It shows how one can have a family of symplectic operators with the help of r-matrices. The chapter shows that the symmetry structure of non-linear integrable system is determined by the bi-Hamiltonian structure. It can be shown that once the symmetry vector fields are known and one has information about the first symplectic structure, one can determine the second symplectic operator. The chapter concludes by presenting yet one method to obtain the Hamiltonian structure from the matrix lax equation, first presented by G. Tu. This method also utilizes the basic properties of a loop algebra and a pivotal role is played by an identity known as the trace identity.
