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We consider Hamiltonian systems with U.1 / symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the absence of linear instability. Essen-tially, the instability is caused by higher algebraic degeneracy of the zero eigen-value in the spectrum of the linearized system. We apply our theory to the non-linear Schrödinger equation. c 2003 Wiley Periodicals, Inc.