On Henri Poincar\'e's note "Sur une forme nouvelle des \'equations de la M\'ecanique"

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.