Variational Derivation of the Equation of Motion of an Ideal Fluid

Continuous Diffeomorphisms

For ideal fluids, the configuration space is the group of volume preserving diffeomorphisms on the fluid container (a region in or that changes with time as a result of the boundary motion). A particle located at a point will travel to a point at time .

The kinetic energy is a mapping from the tangent space to the real numbers


Notice that the Lagrangian satisfies the particle relabeling symmetry expressed as invariance under right composition:

where .

The action is given by

Due to the particle relabeling symmetry the system can be described in terms of a reduced Lagrangian

where is in the Lie algebra of volume preserving diffeomorphisms, and is the identity element of the group.

To obtain the reduced action

Variations must satisfy the Lin Constraint:

The variation in the reduced action is

For a divergence free vector field,


Setting the variation and integrating by parts:

This implies that the integrand must be the gradient of a function

which is equivalent to the Euler equation in coordinate form: