  A class of network models with symmetry group $G$ that evolve as a
Lie-Poisson system is derived from the framework of geometric mechanics, which
generalises the classical Heisenberg model studied in statistical mechanics. We
considered two ways of coupling the spins: one via the momentum and the other
via the position and studied in details the equilibrium solutions and their
corresponding nonlinear stability properties using the energy-Casimir method.
We then took the example $G=SO(3)$ and saw that the momentum-coupled system
reduces to the classical Heisenberg model with massive spins and the
position-coupled case reduces to a new system that has a broken symmetry group
$SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically
observed an interesting synchronisation-like phenomenon for a certain class of
initial conditions. Adding a type of noise and dissipation that preserves the
coadjoint orbit of the network model, we found that the invariant measure is
given by the Gibbs measure, from which the notion of temperature is defined. We
then observed a surprising `triple-humped' phase transition in the heavy
top-like lattice model, where the spins switched from one equilibrium position
to another before losing magnetisation as we increased the temperature. This
work is only a first step towards connecting geometric mechanics with
statistical mechanics and several interesting problems are open for further
investigation.
