  We present in modern language the contents of the famous note published by
Henri Poincar\'e in 1901 "Sur une forme nouvelle des \'equations de la
M\'ecanique", in which he proves that, when a Lie algebra acts locally
transitively on the configuration space of a Lagrangian mechanical system, the
well known Euler-Lagrange equations are equivalent to a new system of
differential equations defined on the product of the configuration space with
the Lie algebra. We write these equations, called the \emph{Euler-Poincar\'e
equations}, under an intrinsic form, without any reference to a particular
system of local coordinates, and prove that they can be conveniently expressed
in terms of the Legendre and momentum maps. We discuss the use of the
Euler-Poincar\'e equation for reduction (a procedure sometimes called
Lagrangian reduction by modern authors), and compare this procedure with the
well known Hamiltonian reduction procedure (formulated in modern terms in 1974
by J.E. Marsden and A. Weinstein). We explain how a break of symmetry in the
phase space produces the appearance of a semi-direct product of groups.
