ランダウ力学 §42問題4 解説

\begin{align} \bm{f} &= \varphi_1 \bm{r} + \varphi_2 \bm{p} + \varphi_3 \bm{r} \times \bm{p} \label{eq_42-ex4-1} \end{align}

\begin{align} \left\{ \bm{f}, M_z \right\} &= \left\{ \varphi_1 \bm{r} + \varphi_2 \bm{p} + \varphi_3 \bm{r} \times \bm{p} , M_z \right\} \nonumber \\ &= \left\{ \varphi_1 \bm{r} , M_z \right\} + \left\{ \varphi_2 \bm{p} , M_z \right\} + \left\{ \varphi_3 \bm{r} \times \bm{p} , M_z \right\} \nonumber \\ &= \left\{ \varphi_1 , M_z \right\} \bm{r} + \left\{ \bm{r} , M_z \right\} \varphi_1 \nonumber \\ &\quad + \left\{ \varphi_2 , M_z \right\} \bm{p} + \left\{ \bm{p} , M_z \right\} \varphi_2 \nonumber \\ &\quad + \left\{ \varphi_3 , M_z \right\} \bm{r} \times \bm{p} + \left\{ \bm{r} \times \bm{p} , M_z \right\} \varphi_3 \nonumber \\ &= \left\{ \bm{r} , M_z \right\} \varphi_1 + \left\{ \bm{p} , M_z \right\} \varphi_2 + \left\{ \bm{M} , M_z \right\} \varphi_3 \nonumber \\ &= \begin{pmatrix} -y \\ x \\ 0 \end{pmatrix} \varphi_1 + \begin{pmatrix} - p_y \\ p_x \\ 0 \end{pmatrix} \varphi_2 + \begin{pmatrix} - M_y \\ M_x \\ 0 \end{pmatrix} \varphi_3 \nonumber \\ &= \begin{pmatrix} - y \varphi_1 - p_y \varphi_2 - M_y \varphi_3 \\ x \varphi_1 + p_x \varphi_2 + M_x \varphi_3 \\ 0 \end{pmatrix} \nonumber \\ &= \begin{pmatrix} - f_y \\ f_x \\ 0 \end{pmatrix} \nonumber \\ &= \bm{f} \times \bm{n} \end{align}