Hamiltonian Methods for Evolution Equations

Abstract

The equilibrium solutions of the equations of nondissipative continuum mechanics are usually found by minimizing an appropriate variational integral. Consequently, smooth solutions will satisfy the Euler-Lagrange equations for the relevant functional and one can employ the group-theoretic methods in the Lagrangian framework discussed in Chapters 4 and 5. However, when presented with the full dynamical problem, one encounters systems of evolution equations for which the Lagrangian viewpoint, even if applicable, no longer is appropriate or natural to the problem. In this case, the Hamiltonian formulation of systems of evolution equations assumes the natural variational role for the system.