It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?

It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the Hamiltonian system is dissipative like a damped harmonic oscillator?

Does the size of an initial volume element in phase space remain constant in time even for dissipative systems?

It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?