Why the phase space fluid is incompressible?

Answer 1

let \rho is density of phase fluid and v is velocity of phase fluid (velocity in a phase space!!)

we take the equation of continuity (we suppose "mass" of phase fluid is conserved) and using Hamilton equations (we suppose classical mechanical system)

[ dot{p_i} = -frac{partial H}{q_i} ]

[ dot{q_i} = +frac{partial H}{p_i} ]

we obtain the result

[ frac{mathrm{d}rho}{mathrm{d}t} = - rho,mathrm{div},v = -rho sum_i left( frac{partialdot{q_i}}{q_i} + frac{partialdot{p_i}}{p_i} right) = 0 ] let \rho is density of phase fluid and v is velocity of phase fluid (velocity in a phase space!!)

we take the equation of continuity (we suppose "mass" of phase fluid is conserved) and using Hamilton equations (we suppose classical mechanical system)

\[ \dot{p_i} = -\frac{\partial H}{q_i} \]

\[ \dot{q_i} = +\frac{\partial H}{p_i} \] we obtain the result

\[ \frac{\mathrm{d}\rho}{\mathrm{d}t} = - \rho\,\mathrm{div}\,v = -\rho \sum_i \left( \frac{\partial\dot{q_i}}{q_i} + \frac{\partial\dot{p_i}}{p_i} \right) = 0 \]