The dynamics of many degrees of freedom interacting via a random potential and subject to a sperical constraint is investigated. Near the dynamical transition the model is described by equations similar to those of the Schematic Mode Coupling Theory of Structural Glasses. Two different universality classes of critical dynamics are found, depending on the nonlinearity of the potential. For both classes we describe the dynamics above and below the transition lines. These results can be relevant for the structural fragile glasses where recent studies of the inherent structures indicate a close analogy between the statistical properties on the N-body potential and random potentials with the same nonlinearities.

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