解析力学⑧ ポアソン括弧
ポアソン括弧とは?
ポアソン括弧の定義とは?
ポアソン括弧
関数$f,g$と正準変数$\{p^i\}$と$\{q^i\}$について、
\begin{align*}
\{f,g\}_{q,p}=\sum_{i=1}^n \left(\dfrac{\partial f}{\partial q^i}\dfrac{\partial g}{\partial q^i}-\dfrac{\partial f}{\partial p^i}\dfrac{\partial g}{\partial q^i}\right)=-\{g,f\}_{q,p}
\end{align*}
ポアソン括弧に関連した式
ひとつめ
関数$f$の時間微分を考えます。\begin{align*}
\dfrac{df}{dt}&=\dfrac{\partial f}{\partial t}+\sum_{i=1}^n \left(\dfrac{\partial f}{\partial q^i}\dfrac{dq^i}{dt}+\dfrac{\partial f}{\partial p^i}\dfrac{dp^i}{dt}\right)
\end{align*}
ここで、もちろん正準変数はハミルトン方程式を満たすので、
\begin{align*}
\dfrac{\partial H}{\partial p^i}&=\dot{q}^i \\
\dfrac{\partial H}{\partial q^i}&=-\dot{p}^i
\end{align*}
という関係があります。よって、
\begin{align*}
\dfrac{df}{dt}&=\dfrac{\partial f}{\partial t}+\sum_{i=1}^n \left(\dfrac{\partial f}{\partial q^i}\dfrac{\partial H}{\partial p^i}-\dfrac{\partial f}{\partial p^i}\dfrac{\partial H}{\partial q^i}\right) \\
&=\dfrac{\partial f}{\partial t}+\{f,H\}_{q,p}
\end{align*}
ふたつめ
\begin{align*}
\{f,p^i\}_{q,p}
&=\dfrac{\partial f}{\partial q^i}\dfrac{\partial p^i}{\partial p^i}-\dfrac{\partial f}{\partial p^i}\dfrac{\partial q^i}{\partial p^i} \\
&=\dfrac{\partial f}{\partial q^i}
\end{align*}
似たような式ですが、
\begin{align*}
\{f,q^i\}_{q,p}
&=\dfrac{\partial f}{\partial q^i}\dfrac{\partial q^i}{\partial p^i}-\dfrac{\partial f}{\partial p^i}\dfrac{\partial q^i}{\partial q^i} \\
&=-\dfrac{\partial f}{\partial p^i}
\end{align*}