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

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