classical mechanics - What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam:

The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov function for controlling the position and velocity of the system. Consider a damped single degree-of-freedom system, $m\ddot{x}+c\dot{x}+kx=0$, where $m$ is the mass, $c$ is the velocity-proportional damping and $k$ is the stiffness. A candidate Lyapunov function is the Hamiltonian $V=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2$. What are the reasons for leaving out the dissipative energy term when writing the Lyapunov function?

The only thing what comes into my mind for this question is, that a dissipative energy term in the Lyapunov function would have a "-" sign and the Lyapunov function would thus not be positive definite anymore. Is that correct?