Classical Liouville Equation

Available identifiers

WikidataQ766722 ScholiaQ766722MaRDI QIDQ6674267

partial differential equation for the that time rate of change of density of points in phase space

Description

Consider a Hamiltonian dynamical system with canonical coordinates and conjugate momenta. Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume.
Note the similarity with the quantum Liouville (von Neumann) equation where the Poisson brackets {.,.} are replaced by commutator brackets [.,.]

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Defining Formula: $\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum _{i=1}^n(\frac{\partial \rho }{\partial q_i}{\dot{q}}_i+\frac{\partial \rho }{\partial p_i}{\dot{p}}_i)$

$H$ symbol represents Classical Hamilton Function

$\rho$ symbol represents Classical Density (Phase Space)

$p$ symbol represents Classical Momentum

$q$ symbol represents Classical Position

$t$ symbol represents Time

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