ABSTRACT
This chapter presents two-dimensional Euler equations for a homogeneous fluid. There is a long history in connection with the study of the existence, uniqueness and regularity of solutions to the Cauchy problem associated with the two-dimensional incompressible Euler equations. In many situations of interest, a system of partial differential equations does not have a least action principle associated with it in terms of the desired dependent variables. If a given system of partial differential equations is associated with a physical problem for which there exists a canonical Hamiltonian description written in terms of the canonical positions and momenta, it may be possible to introduce a sequence of transformations into the canonical Poisson bracket and systematically derive the Poisson bracket as a function of the desired noncanonical dependent variables. The connection between symmetry or invariance properties of the Hamiltonian structure and a particular conserved functional is expressed through Noether’s Theorem.
