  We survey physical models which capture the main concepts of double field
theory on para-Hermitian manifolds. We show that the geometric theory of
Lagrangian and Hamiltonian dynamical systems is an instance of para-Kahler
geometry which extends to a natural example of a Born geometry. The
corresponding phase space geometry belongs to the family of natural almost
para-Kahler structures which we construct explicitly as deformations of the
canonical para-Kahler structure by non-linear connections. We extend this
framework to a class of non-Lagrangian dynamical systems which naturally
encodes the notion of fluxes in para-Hermitian geometry. In this case we
describe the emergence of fluxes in terms of weak integrability defined by the
D-bracket, and we extend the construction to arbitrary cotangent bundles where
we reproduce the standard generalized fluxes of double field theory. We also
describe the para-Hermitian geometry of Drinfel'd doubles, which gives an
explicit illustration of the interplay between fluxes, D-brackets and different
polarizations. The left-invariant para-Hermitian structure on a Drinfel'd
double in a Manin triple polarization descends to a doubled twisted torus,
which we use to illustrate how changes of polarizations give rise to different
fluxes and string backgrounds in para-Hermitian geometry.
