  We perform a microcanonical study of classical lattice phi^4 field models in
3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these
systems that undergo a second order phase transition in the thermodynamic limit
are here investigated. The microscopic Hamiltonian dynamics neatly reveals the
presence of a phase transition through the time averages of conventional
thermodynamical observables. Moreover, peculiar behaviors of the largest
Lyapunov exponents at the transition point are observed. A Riemannian
geometrization of Hamiltonian dynamics is then used to introduce other relevant
observables, that are measured as functions of both energy density and
temperature. On the basis of a simple and abstract geometric model, we suggest
that the apparently singular behaviour of these geometric observables might
probe a major topological change of the manifolds whose geodesics are the
natural motions.
