  This paper reports a development in the proper symplectic decomposition (PSD)
for model reduction of forced Hamiltonian systems. As an analogy to the proper
orthogonal decomposition (POD), PSD is designed to build a symplectic subspace
to fit empirical data. Our aim is two-fold. First, to achieve computational
savings for large-scale Hamiltonian systems with external forces. Second, to
simultaneously preserve the symplectic structure and the forced structure of
the original system. We first reformulate d'Alembert's principle in the
Hamiltonian form. Corresponding to the integral and local forms of d'Alembert's
principle, we propose two different structure-preserving model reduction
approaches to reconstruct low-dimensional systems, based on the variational
principle and on the structure-preserving projection, respectively. These two
approaches are proven to yield the same reduced system. Moreover, by
incorporating the vector field into the data ensemble, we provided several
algorithms for energy preservation. In a special case when the external force
is described by the Rayleigh dissipative function, the proposed method
automatically preserves the dissipativity, boundedness, and stability of the
original system. The stability, accuracy, and efficiency of the proposed method
are illustrated through numerical simulations of a dissipative wave equation.
