  Manifolds with boundary, with corners, $b$-manifolds and foliations model
configuration spaces for particles moving under constraints and can be
described as $E$-manifolds. $E$-manifolds were introduced in [NT01] and
investigated in depth in [MS20]. In this article we explore their physical
facets by extending gauge theories to the $E$-category. Singularities in the
configuration space of a classical particle can be described in several new
scenarios unveiling their Hamiltonian aspects on an $E$-symplectic manifold.
Following the scheme inaugurated in [Wei78], we show the existence of a
universal model for a particle interacting with an $E$-gauge field. In
addition, we generalize the description of phase spaces in Yang-Mills theory as
Poisson manifolds and their minimal coupling procedure, as shown in [Mon86],
for base manifolds endowed with an $E$-structure. In particular, the reduction
at coadjoint orbits and the shifting trick are extended to this framework. We
show that Wong's equations, which describe the interaction of a particle with a
Yang-Mills field, become Hamiltonian in the $E$-setting. We formulate the
electromagnetic gauge in a Minkowski space relating it to the proper time
foliation and we see that our main theorem describes the minimal coupling in
physical models such as the compactified black hole.
