Ecole polytechnique federale de Lausanne,

EPFL-DMA, CH-1015 Lausanne,

E-mail: oliver.maspfuhl@epfl.ch

**Linearized Poisson geometry and gauge fields**

**Abstract:**

In the paper, we show how classical dynamics of particles in a gravitational
and Yang-Mills field emerges naturally from the geometry of a general Poisson
manifold as a second order approximation of a Hamiltonian system on this
manifold. The Hamiltonian only has to have vanishing differential on some
Lagrangian submanifold *X* of a locally minimal, polarized symplectic
leaf and satisfy a non-degeneracy condition. Furthermore, Higgs fields
are naturally present if the systems in coisotropically constraint. The
most important feature of the work is the definition of an *E*-connection
form associated to such a Hamiltonian on *X*, where *E* is a
natural Lie algebroid over *X*.