Variational Principles in Fluid Dynamics, Symmetries and Conservation Laws

Abstract

In this chapter, the Lagrangian density associated with fluid dynamics will be introduced. The equations of motion will be rederived from the Lagrangian density using Hamilton’s principle. In particular, Hamilton’s principle will be applied mainly in the Lagrangian framework, where the analogy to a system of point particles will simplify the calculations. The same principle will, however, be applied also in the Eulerian framework, and the relationship between the two frameworks will be revealed from the use of canonical transformations. Noether’s Theorem will be applied to derive the conservation laws corresponding to the continuous symmetries of the Lagrangian density for the ideal fluid. Particular attention will be given to the particle relabelling symmetry and the associated conservation of vorticity.