ラグランジアンとガリレイ変換

ラグランジアンとガリレイ変換

$$\begin{array}{rcl}L& =& \frac{m}{2}{\dot{q}}^2\end{array}$$ $$\begin{array}{rcl}{q}^{\prime }& =& q+Vt\dots \mathrm{系}\mathrm{が}V(\mathrm{一}\mathrm{定}\mathrm{値})\mathrm{で}\mathrm{移}\mathrm{動}\mathrm{し}\mathrm{て}\mathrm{い}\mathrm{る}\\ \dot{q^{\prime }}& =& \dot{q}+V\dots \frac{\mathrm{d}}{\mathrm{d}t}(q+Vt)=\frac{\mathrm{d}}{\mathrm{d}t}q+\frac{\mathrm{d}}{\mathrm{d}t}Vt\\ L^{\prime }& =& \frac{m}{2}(\dot{q^{\prime }}{)}^2\\ & =& \frac{m}{2}(\dot{q}+V{)}^2\\ & =& \frac{m}{2}({\dot{q}}^2+2\dot{q}V+V^2)\\ & =& \frac{m}{2}{\dot{q}}^2+\frac{m}{2}(2\dot{q}V+V^2)\\ & =& L+\frac{m}{2}(2\dot{q}V+V^2)\\ \frac{\partial L^{\prime }}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L^{\prime }}{\partial \dot{q}}& =& \{\frac{\partial L}{\partial q}+\frac{\partial }{\partial q}(2\dot{q}V+V^2)\}-[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}\{\frac{m}{2}(2\dot{q}V+V^2)\}]\\ & =& \frac{\partial L}{\partial q}-(\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}+\frac{\mathrm{d}}{\mathrm{d}t}mV)\dots \frac{\mathrm{d}}{\mathrm{d}q}(2\dot{q}V+V^2)=0\dots q\mathrm{と}\dot{q}\mathrm{は}\mathrm{独}\mathrm{立}\\ & =& \frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\dots \frac{\mathrm{d}}{\mathrm{d}t}mV=0\dots \mathrm{こ}\mathrm{こ}\mathrm{で}\mathrm{は}V\mathrm{は}\mathrm{定}\mathrm{数}\mathrm{で}\mathrm{あ}\mathrm{る}\end{array}$$