  Topological constraints on a dynamical system often manifest themselves as
breaking of the Hamiltonian structure; well-known examples are non-holonomic
constraints on Lagrangian mechanics. The statistical mechanics under such
topological constraints is the subject of the present study. Conventional
arguments based on phase spaces, Jacobi identity, invariant measure, or the H
theorem are no longer applicable, since all these notions stem from the
symplectic geometry underlying canonical Hamiltonian systems. Remembering that
Hamiltonian systems are endowed with field tensors (canonical 2-forms) that
have zero helicity, our mission is to extend the scope toward the class of
systems governed by finite-helicity field tensors. Here we introduce a new
class of field tensors that are characterized by Beltrami vectors. We prove an
H theorem for this Beltrami class. The most general class of energy-conserving
systems are non-Beltrami, for which we identify the "field charge" that
prevents the entropy to maximize, resulting in creation of heterogeneous
distributions. The essence of the theory can be delineated by classifying
three-dimensional dynamics. We then generalize to arbitrary (finite)
dimensions.
