式展開: 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{array}{rclrc} L^{'} & = & L + \frac{d}{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{d}{d t} \frac{\partial L^{'}}{\partial \dot{q}} & = & (\frac{\partial L}{\partial q} + \frac{\partial}{\partial q} \frac{d}{d t} f) - (\frac{d}{d t} \frac{\partial L^{'}}{\partial \dot{q}} + \frac{d}{d t} \frac{\partial}{\partial \dot{q}} \frac{d}{d t} f) \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + (\frac{\partial}{\partial q} \frac{d}{d t} f - \frac{d}{d t} \frac{\partial}{\partial \dot{q}} \frac{d}{d t} f) \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + {\frac{\partial}{\partial q} (\frac{\partial f}{\partial q} \dot{q} + \frac{\partial f}{\partial t}) - \frac{d}{d t} \frac{\partial}{\partial \dot{q}} (\frac{\partial f}{\partial q} \dot{q} + \frac{\partial f}{\partial t})} \dots \frac{d}{d t} f (q, t) & = & \frac{\partial f (q, t)}{\partial q} \frac{d q}{d t} + \frac{\partial f (q, t)}{\partial t} \frac{d t}{d t} \\ = & \frac{\partial f (q, t)}{\partial q} \frac{d q}{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} \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + {\frac{\partial}{\partial q} (\frac{\partial f}{\partial q} \dot{q} + \frac{\partial f}{\partial t}) - \frac{d}{d t} \frac{\partial f}{\partial q}} \dots \frac{\partial}{\partial \dot{q}} (\frac{\partial f (q, t)}{\partial q} \dot{q} + \frac{\partial f (q, t)}{\partial t}) & = & \frac{\partial f (q, t)}{\partial q} \dots q と \dot{q} は 独 立 \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + {\frac{\partial}{\partial q} (\frac{\partial f}{\partial q} \dot{q} + \frac{\partial f}{\partial t}) - (\frac{\partial}{\partial q} \frac{\partial f}{\partial q} \dot{q} + \frac{\partial}{\partial t} \frac{\partial f}{\partial q})} \dots \frac{d}{d t} \frac{\partial f (q, t)}{\partial q} & = & \frac{\partial}{\partial q} \frac{\partial f (q, t)}{\partial q} \frac{d q}{d t} + \frac{\partial}{\partial t} \frac{\partial f (q, t)}{\partial q} \frac{d t}{d t} \\ = & \frac{\partial}{\partial q} \frac{\partial f (q, t)}{\partial q} \frac{d q}{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} \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + {\frac{\partial}{\partial q} \frac{\partial f}{\partial q} \dot{q} + \frac{\partial}{\partial q} \frac{\partial f}{\partial t} - (\frac{\partial}{\partial q} \frac{\partial f}{\partial q} \dot{q} + \frac{\partial}{\partial t} \frac{\partial f}{\partial q})} \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + (\frac{\partial}{\partial q} \frac{\partial f}{\partial t} - \frac{\partial}{\partial t} \frac{\partial f}{\partial q}) \\ = & (\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}}) + (\frac{\partial}{\partial q} \frac{\partial f}{\partial t} - \frac{\partial}{\partial q} \frac{\partial f}{\partial t}) \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{d}{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{array}

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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

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