  We consider Lagrangians in Hamilton's principle defined on the tangent space
$TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of
$G$ leads to the symmetry-reduced Euler-Lagrange equations called the
Euler-Poincar\'e equations. In this case, the invariant Lagrangian is defined
on the Lie algebra of the group and its Euler-Poincar\'e equations are defined
on the dual Lie algebra, where dual is defined by the operation of taking
variational derivative. On the Hamiltonian side, the Euler-Poincar\'e equations
are Lie-Poisson and they possess accompanying momentum maps, which encode both
their conservation laws and the geometry of their solution space.
  The standard Euler-Poincar\'e examples are treated, including particle
dynamics, the rigid body, the heavy top and geodesic motion on Lie groups.
Additional topics deal with Fermat's principle, the $\mathbb{R}^3$ Poisson
bracket, polarized optical traveling waves, deformable bodies (Riemann
ellipsoids) and shallow water waves, including the integrable shallow water
wave systems associated with geodesic motion on the diffeomorphisms. The
lectures end with the semidirect-product Euler-Poincar\'e reduction theorem for
ideal fluid dynamics. This theorem introduces the Euler--Poincar\'e variational
principle for incompressible and compressible motions of ideal fluids, with
applications to geophysical fluids. It also leads to their Lie-Poisson
Hamiltonian formulation.
