  We consider model order reduction of parameterized Hamiltonian systems
describing nondissipative phenomena, like wave-type and transport dominated
problems. The development of reduced basis methods for such models is
challenged by two main factors: the rich geometric structure encoding the
physical and stability properties of the dynamics and its local low-rank
nature. To address these aspects, we propose a nonlinear structure-preserving
model reduction where the reduced phase space evolves in time. In the spirit of
dynamical low-rank approximation, the reduced dynamics is obtained by a
symplectic projection of the Hamiltonian vector field onto the tangent space of
the approximation manifold at each reduced state. A priori error estimates are
established in terms of the projection error of the full model solution onto
the reduced manifold. For the temporal discretization of the reduced dynamics
we employ splitting techniques. The reduced basis satisfies an evolution
equation on the manifold of symplectic and orthogonal rectangular matrices
having one dimension equal to the size of the full model. We recast the problem
on the tangent space of the matrix manifold and develop intrinsic temporal
integrators based on Lie group techniques together with explicit Runge-Kutta
(RK) schemes. The resulting methods are shown to converge with the order of the
RK integrator and their computational complexity depends only linearly on the
dimension of the full model, provided the evaluation of the reduced flow
velocity has a comparable cost.
