The Liouville Equation for Hamiltonian Systems

Until now we have considered a general system z = /(z). What happens when/ is derived from a Hamiltonian  

From the above we know that for a smooth scalar function u = u(z), 

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To date, there has only been one attempt to develop a dynamic density functional theory for systems in which inertia plays a role [8]. However, it has been shown that the formal proof for the existence of a quantum mechanical dynamical density functional theory by Runge and Gross can be applied to classical systems [9] by starting from the Liouville equation for Hamiltonian systems (instead of the time-dependent Schrodinger equation), which therefore includes inertia terms. However, the proof is not of practical use (see below). 

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