Realizations of Lie Algebras in Classical Mechanics

Classical Poisson bracket realizations of semisimple Lie algebras are considered. An attempt is made to determine the minimum number of canonical degrees of freedom needed to find a realization of a given Lie algebra. Under the restriction to the symmetric traceless tensor representations of the orthogonal groups, and the symmetric tensor representations of the unimodular unitary groups, it is shown that with n pairs of canonical variables one can find realizations of the Lie algebras of O(n + 2) and SU(n + 1), but no higher groups.