Real hamiltonian logarithm of a symplectic matrix

AbstractIn this note we give sharp conditions under which a real symptectic matrix S has a real Hamiltonian logarithm, and explicitly construct a logarithm. In the classical work of Williamson, see (J. Williamson, Amer. J. Math. 61 (1939) 897–911), necessary and sufficient conditions were already given. Our contribution is to provide constructive arguments based on the canonical form of real symplectic matrices derived by Laub and Meyer (A.J. Laub and K.R. Mcyer, J. Celestial mechanics, 9 (1974) 213–238)