  A general framework for the numerical approximation of evolution problems is
presented that allows to preserve exactly an underlying Hamiltonian- or
gradient structure. The approach relies on rewriting the evolution problem in a
particular form that complies with the underlying geometric structure. The
Galerkin approximation of a corresponding variational formulation in space then
automatically preserves this structure which allows to deduce important
properties for appropriate discretization schemes including projection based
model order reduction. We further show that the underlying structure is
preserved also under time discretization by a Petrov-Galerkin approach. The
presented framework is rather general and allows the numerical approximation of
a wide range of applications, including nonlinear partial differential
equations and port-Hamiltonian systems. Some examples will be discussed for
illustration of our theoretical results and connections to other discretization
approaches will be revealed.
