It is shown that, under general circumstances, symplectic G-orbitsin a hamiltonian manifold acted on (symplectically) by a Lie group G provide critical points for the norm squared of the moment map. This fact yields a “variational” interpretation of the symplectic orbits appearing in the projective space attached to an irreducible representation of a compact simple Lie group (according to work ofKostant and Sternbergand of Giavarini and Onofri),where the previous function is also related to the invariant uncertainty considered by Delbourgo and Perelomov. A notion of generalized canonical conjugate variables (in the Kithler case) is also presented and used in the framework of a KAhIer geometric interpretation of the Heisenberg uncertainty relations (building on the analysis given by Cirelli, Mania and Pizzocchero and by Provost and Vallee); it is proved, in particular, that the generalized coherent states of Rawnsley minimize the uncertainty relationsfor any pair of generalized canonically conjugate variables.
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