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
where
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,
Therefore,
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:
