  These are lecture notes for a short winter course at the Department of
Mathematics, University of Coimbra, Portugal, December 6--8, 2018. The course
was part of the 13th International Young Researchers Workshop on Geometry,
Mechanics and Control.
  In three lectures I trace the work of three heroes of mathematics and
mechanics: Euler, Poincar\'e, and Arnold. This leads up to the aim of the
lectures: to explain Arnold's discovery from 1966 that solutions to Euler's
equations for the motion of an incompressible fluid correspond to geodesics on
the infinite-dimensional Riemannian manifold of volume preserving
diffeomorphisms. In many ways, this discovery is the foundation for the field
of geometric hydrodynamics, which today encompasses much more than just Euler's
equations, with deep connections to many other fields such as optimal
transport, shape analysis, and information theory.
