  We study a pair of infinite dimensional dynamical systems naturally
associated with the study of minimizing/maximizing functions for the Strichartz
inequalities for the Schr\"odinger equation. One system is of gradient type and
the other one is a Hamiltonian system. For both systems, the corresponding sets
of critical points, their stability, and the relation between the two are
investigated. By a combination of numerical and analytical methods we argue
that the Gaussian is a maximizer in a class of Strichartz inequalities for
dimensions one, two and three. The argument reduces to verification of an
apparently new combinatorial inequality involving binomial coefficients.
