ハミルトン力学
A ̇ = q ̇ i ∂ A ∂ q i + ∂ A ∂ p i p ̇ i + ∂ A ∂ t = ∂ H ∂ p i ∂ A ∂ q i − ∂ A ∂ p i ∂ H ∂ q i + ∂ A ∂ t {\displaystyle {\dot {A}}={\dot {q}}_{i}\,{\frac {\partial A}{\partial q_{i}}}+{\frac {\partial A}{\partial p_{i}}}{\dot {p}}_{i}+{\frac {\partial A}{\partial t}}={\frac {\partial H}{\partial p_{i}}}{\frac {\partial A}{\partial q_{i}}}-{\frac {\partial A}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}+{\frac {\partial A}{\partial t}}}
となる。特にハミルトニアンの時間微分は
H ̇ = ∂ H ∂ p i ∂ H ∂ q i − ∂ H ∂ p i ∂ H ∂ q i + ∂ H ∂ t = ∂ H ∂ t {\displaystyle {\dot {H}}={\frac {\partial H}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}-{\frac {\partial H}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}+{\frac {\partial H}{\partial t}}={\frac {\partial H}{\partial t}}}
である。