In this paper, we propose a constructive procedure to modify the Hamiltonia
n function of forced Hamiltonian systems with dissipation in order to gener
ate Lyapunov functions for nonzero equilibria, A key step in the procedure,
which is motivated from energy-balance considerations standard in network
modeling of physical systems, is to embed the system into a larger Hamilton
ian system for which a series of Casimir functions can be easily constructe
d. Interestingly enough, for linear systems the resulting Lyapunov function
is the incremental energy; thus our derivations provide a physical explana
tion to it. An easily verifiable necessary and sufficient condition for the
applicability of the technique in the general nonlinear case is given. Som
e examples that illustrate the method are given.