ランダウ・リフシッツ力学 §9 問題 2

ランダウ・リフシッツの力学(増補第3版) §9 問題 2 のメモです。 \begin{align} M_x = & m (y\dot{z} – z \dot{y}) \\ = & m \left[ r\sin\varphi \sin \theta (\dot{r} \cos\theta -r\dot{\theta} \sin\theta) – r\cos\theta(\dot{r}\sin\varphi \sin\theta + r\dot{\varphi}\cos\varphi \sin\theta + r\sin\varphi \dot{\theta} \cos \theta ) \right] \\ = & – mr^2 (\dot{\theta}\sin\varphi + \dot{\varphi} \sin\theta \cos\theta\cos\varphi)\\ M_y = & m(z\dot{x} – x \dot{z}) \\ = & m r\cos \theta \left( \dot{r} \cos\varphi\sin\theta – r\dot{\varphi} \sin\varphi\sin\theta + r\dot{\theta} \cos\varphi \cos\theta \right) – mr\cos\varphi \sin\theta(\dot{r} \cos\theta – r \dot{\theta} \sin\theta ) \\ = & m r^2 ( \dot{\theta} \cos\varphi – \dot{\varphi} \cos\theta \sin\theta \sin\varphi) \\ M_z = & m (x\dot{y} – y \dot{x}) \\ = & m r \cos\varphi \sin\theta(\dot{r} \sin\varphi\sin\theta + r\dot{\varphi} \cos\varphi\sin\theta + r\dot{\theta} \sin\varphi \cos\theta ) \\ & – mr \sin\varphi \sin \theta (\dot{r} \cos\varphi \sin\theta – r \dot{\varphi} \sin\varphi \sin\theta + r \dot{\theta} \cos\varphi \cos\theta ) \\ = & m r^2 \dot{\varphi} \sin^2 \theta \\ M^2 = & m^2 r^4 (\dot{\theta}\sin\varphi + \dot{\varphi} \sin\theta \cos\theta\cos\varphi)^2 \\ & + m^2 r^4 ( \dot{\theta} \cos\varphi – \dot{\varphi} \cos\theta \sin\theta \sin\varphi)^2 \\ & + m^2 r^4 \dot{\varphi}^2 \sin^4 \theta \\ = & m^2 r^4 \dot{\theta}^2 + m^2 r^4 \dot{\varphi}^2 \sin^2\theta \cos^2\theta + m^2 r^4 \dot{\varphi}^2 \sin^4\theta \\ = & m^2 r^4 \dot{\theta}^2 + m^2 r^4 \dot{\varphi}^2 \sin^2\theta \end{align}