Lagrangian & Hamiltonian Dynamics

Let 𝜌(p , q , t) be the phase space density of an ensemble of a system. The Hamiltonian of the system is H(p , q ). If {A, B} denotes the Poisson bracket of A and B, then d𝜌 dt = 0 implies

Options:

• (A) 𝜕𝜌 𝜕t = 0
• (B) 𝜕𝜌 𝜕t ∝ {𝜌, H}
• (C) 𝜕𝜌 𝜕t ∝ {𝜌, p ∙ q 2 }
• (D) 𝜕𝜌 𝜕t ∝ {𝜌, q ∙ q 2 }