リウヴィルの定理 (物理学)
∂ ρ ∂ t + ∑ i = 1 n ( ∂ ( ρ q ̇ i ) ∂ q i + ∂ ( ρ p ̇ i ) ∂ p i ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{n}\left({\frac {\partial (\rho {\dot {q}}_{i})}{\partial q_{i}}}+{\frac {\partial (\rho {\dot {p}}_{i})}{\partial p_{i}}}\right)=0}
に基づいている。
すなわち、三つ組 ( ρ , ρ q ̇ i , ρ p ̇ i ) {\displaystyle (\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})} は保存カレント(英語版)である。リウヴィル方程式と項
ρ ∑ i = 1 n ( ∂ q ̇ i ∂ q i + ∂ p ̇ i ∂ p i ) = ρ ∑ i = 1 n ( ∂ 2 H ∂ q i ∂ p i − ∂ 2 H ∂ p i ∂ q i ) = 0 , {\displaystyle \rho \sum _{i=1}^{n}\left({\frac {\partial {\dot {q}}_{i}}{\partial q_{i}}}+{\frac {\partial {\dot {p}}_{i}}{\partial p_{i}}}\right)=\rho \sum _{i=1}^{n}\left({\frac {\partial ^{2}H}{\partial q_{i}\,\partial p_{i}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q_{i}}}\right)=0,}