  The motion of a classical particle in a gravitational and a Yang-Mills field
was described by S. Sternberg and A. Weinstein by a particular Hamiltonian
system on a Poisson manifold known under the name of Sternberg-Weinstein phase
space. This system leads to the generalization of the Lorentz equation of
motion first discovered by Wong. The aim of this work is to show that
inversely, a Hamiltonian H on a general Poisson manifold, with the property
that its differential vanishes on a Lagrangian submanifold X of a symplectic
leaf and is generic in any other direction, naturally defines a metric on X, as
well as a principal connection form on a canonical principal fiber bundle on X.
These fields, which are credited to model a gravitational and a Yang-Mills
field on X, respectively, define a linearized Hamiltonian system of Wong type
on a canonical linearized Poisson manifold at X locally isomorphic to a
Sternberg-Weinstein phase space. In addition, H is shown to define scalar
fields which first appeared in a theory of Einstein and Mayer. In the presence
of a coisotropic constraint, the reduced system can be regarded as the phase
space of particles in gravitational, Yang-Mills and Higgs fields. We further
show that all our constructions are locally related to usual gauge and
Kaluza-Klein theory via symplectic realization.
