ランダウ力学 §42問題2 解説

\begin{align} \left\{ x , M_x \right\} &= \left\{ x , y p_z - z p_y \right\} \nonumber \\ &= \left\{ x , y p_z \right\} - \left\{ x , z p_y \right\} \nonumber \\ &= \left\{ x , y \right\} p_z + y \left\{ x , p_z \right\} - \left\{ x , z \right\} p_y - z \left\{ x , p_y \right\} \nonumber \\ &= 0 \nonumber \end{align}

\begin{align} \left\{ x , M_y \right\} &= \left\{ x , z p_x - x p_z \right\} \nonumber \\ &= \left\{ x , z p_x \right\} - \left\{ x , x p_z \right\} \nonumber \\ &= \left\{ x , z \right\} p_x + z \left\{ x , p_x \right\} - \left\{ x , x \right\} p_z - x \left\{ x , p_z \right\} \nonumber \\ &= z \nonumber \end{align} であり, \begin{align} \left\{ x , M_z \right\} &= \left\{ x , x p_y - y p_x \right\} \nonumber \\ &= \left\{ x , x p_y \right\} - \left\{ x , y p_x \right\} \nonumber \\ &= \left\{ x , x \right\} p_y + x \left\{ x , p_y \right\} - \left\{ x , y \right\} p_x - y \left\{ x , p_x \right\} \nonumber \\ &= - y \nonumber \end{align}

\begin{align} \left\{ M_x , M_x \right\} &= 0 \quad (\because \text{歪対称性}) \nonumber \end{align}

\begin{align} \left\{ M_x , M_y \right\} &= \left\{ y p_z - z p_y , M_y \right\} \nonumber \\ &= \left\{ y p_z , M_y \right\} - \left\{ z p_y , M_y \right\} \nonumber \\ &= y \left\{ p_z , M_y \right\} + \left\{ y , M_y \right\} p_z - z \left\{ p_y , M_y \right\} - \left\{ z , M_y \right\} p_y \nonumber \\ &= y \left( - p_x \right) + 0 \cdot p_z - z \cdot 0 -\left( - x \right) p_y \nonumber \\ &= x p_y - y p_x \nonumber \\ &= M_z \nonumber \end{align}

\begin{align} \left\{ M_x , M_z \right\} &= \left\{ y p_z - z p_y , M_z \right\} \nonumber \\ &= \left\{ y p_z , M_z \right\} - \left\{ z p_y , M_z \right\} \nonumber \\ &= y \left\{ p_z , M_z \right\} + \left\{ y , M_z \right\} p_z - z \left\{ p_y , M_z \right\} - \left\{ z , M_z \right\} p_y \nonumber \\ &= y \cdot 0 + x p_z - z p_x - 0 \cdot p_y \nonumber \\ &= - \left( z p_x - x p_z \right) \nonumber \\ &= - M_y \nonumber \end{align}