ラグランジアンを変えない加えてもよい凾数

ラグランジアンを変えない加えてもよい凾数

$$\begin{eqnarray} L'&=&L+\frac{\mathrm{d}}{\mathrm{d}t} f(q, t)\;\dots\;任意凾数f(q,t)の時間に関する完全導凾数(全微分)\\ \frac{\partial L'}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L'}{\partial \dot{q}} &=&\left ( \frac{\partial L}{\partial q} + \frac{\partial }{\partial q}\frac{\mathrm{d}}{\mathrm{d}t} f \right ) - \left ( \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L'}{\partial \dot{q}} + \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}\frac{\mathrm{d}}{\mathrm{d}t} f \right )\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left( \frac{\partial }{\partial q}\frac{\mathrm{d}}{\mathrm{d}t} f - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}\frac{\mathrm{d}}{\mathrm{d}t} f \right)\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left\{ \frac{\partial }{\partial q}\left( \frac{\partial f}{\partial q}\dot{q} + \frac{\partial f}{\partial t} \right) - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}\left( \frac{\partial f}{\partial q}\dot{q} + \frac{\partial f}{\partial t} \right) \right\} \;\dots\; \frac{\mathrm{d}}{\mathrm{d}t} f(q, t) &=&\frac{\partial f(q,t)}{\partial q}\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{\partial f(q,t)}{\partial t}\frac{\mathrm{d}t}{\mathrm{d}t}\\ &&&=&\frac{\partial f(q,t)}{\partial q}\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{\partial f(q,t)}{\partial t}\\ &&&=&\frac{\partial f(q,t)}{\partial q}\dot{q} + \frac{\partial f(q,t)}{\partial t}\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left\{ \frac{\partial }{\partial q}\left( \frac{\partial f}{\partial q}\dot{q} + \frac{\partial f}{\partial t} \right) - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial f}{\partial q} \right\} \;\dots\; \frac{\partial}{\partial \dot{q}}\left( \frac{\partial f(q,t)}{\partial q}\dot{q} + \frac{\partial f(q,t)}{\partial t} \right)&=&\frac{\partial f(q,t)}{\partial q}\;\dots\;qと\dot{q}は独立\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left\{ \frac{\partial }{\partial q}\left( \frac{\partial f}{\partial q}\dot{q} + \frac{\partial f}{\partial t} \right) - \left(\frac{\partial}{\partial q}\frac{\partial f}{\partial q}\dot{q} + \frac{\partial}{\partial t}\frac{\partial f}{\partial q}\right) \right\} \;\dots\; \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial f(q,t)}{\partial q} &=&\frac{\partial}{\partial q}\frac{\partial f(q,t)}{\partial q}\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{\partial}{\partial t}\frac{\partial f(q,t)}{\partial q}\frac{\mathrm{d}t}{\mathrm{d}t}\\ &&&=&\frac{\partial}{\partial q}\frac{\partial f(q,t)}{\partial q}\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{\partial}{\partial t}\frac{\partial f(q,t)}{\partial q}\\ &&&=&\frac{\partial}{\partial q}\frac{\partial f(q,t)}{\partial q}\dot{q} + \frac{\partial}{\partial t}\frac{\partial f(q,t)}{\partial q}\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left\{ \frac{\partial }{\partial q}\frac{\partial f}{\partial q}\dot{q} +\frac{\partial }{\partial q} \frac{\partial f}{\partial t} - \left(\frac{\partial}{\partial q}\frac{\partial f}{\partial q}\dot{q} + \frac{\partial}{\partial t}\frac{\partial f}{\partial q}\right) \right\}\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left( \frac{\partial }{\partial q} \frac{\partial f}{\partial t} - \frac{\partial}{\partial t}\frac{\partial f}{\partial q} \right)\\ &=&\left( \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \right) + \left( \frac{\partial }{\partial q} \frac{\partial f}{\partial t} - \frac{\partial}{\partial q}\frac{\partial f}{\partial t} \right)\;\dots\;偏微分の順番の入れ替え条件を満たす場合\\ &=&\frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} \;\dots\;L'で立てた方程式はLで立て方程式と等しくなる \end{eqnarray}$$