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}{rcl}{L}^{\prime }& =& L+\frac{\mathrm{d}}{\mathrm{d}t}f(q,t)\dots \mathrm{任}\mathrm{意}\mathrm{凾}\mathrm{数}f(q,t)\mathrm{の}\mathrm{時}\mathrm{間}\mathrm{に}\mathrm{関}\mathrm{す}\mathrm{る}\mathrm{完}\mathrm{全}\mathrm{導}\mathrm{凾}\mathrm{数}(\mathrm{全}\mathrm{微}\mathrm{分})\\ \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}\frac{\mathrm{d}}{\mathrm{d}t}f)-(\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L^{\prime }}{\partial \dot{q}}+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}\frac{\mathrm{d}}{\mathrm{d}t}f)\\ & =& (\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}})+(\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)\\ & =& (\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{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{\mathrm{d}}{\mathrm{d}t}\frac{\partial }{\partial \dot{q}}(\frac{\partial f}{\partial q}\dot{q}+\frac{\partial f}{\partial t})\}\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}\\ & =& (\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{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{\mathrm{d}}{\mathrm{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\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}})+\{\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{\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}\\ & =& (\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{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{\mathrm{d}}{\mathrm{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{\mathrm{d}}{\mathrm{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 \mathrm{偏}\mathrm{微}\mathrm{分}\mathrm{の}\mathrm{順}\mathrm{番}\mathrm{の}\mathrm{入}\mathrm{れ}\mathrm{替}\mathrm{え}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{場}\mathrm{合}\\ & =& \frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\dots L^{\prime }\mathrm{で}\mathrm{立}\mathrm{て}\mathrm{た}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{は}L\mathrm{で}\mathrm{立}\mathrm{て}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{と}\mathrm{等}\mathrm{し}\mathrm{く}\mathrm{な}\mathrm{る}\end{array}$$

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

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

ラグランジュ方程式

ラグランジュ方程式

位置$q$, 速度$\dot{q}$($=\frac{\mathrm{d}q}{\mathrm{d}t}$), 系を特徴付ける凾数(ラグランジアン)$L(q,\dot{q},t)$とすると $L$の時間$t$での積分である作用$S$が最小になるように系は運動する(最小作用の原理/ハミルトンの原理)． $$\begin{array}{rcl}S& =& {\int }_{t_1}^{t_2}L(q,\dot{q},t)\mathrm{d}t\end{array}$$ よって$S$が最小となるような$q(t)$が系の運動である．
$q(t)$から微小量$\delta q(t)$ずれた$q(t)+\delta q(t)$を通る$L(q+\delta q,\dot{q}+\delta \dot{q},t)$では，$S$は増加する． $q(t)$を通る時との差を$\delta S$とする．
(ただし$\delta q(t)$は$\delta q(t_1)=\delta q(t_2)=0$とし，始点と終点では，ずれはないものとする(境界条件を満たす)) $$\begin{array}{rcl}\delta S& \equiv & {\int }_{t_1}^{t_2}L(q+\delta q,\dot{q}+\delta \dot{q},t)\mathrm{d}t-{\int }_{t_1}^{t_2}L(q,\dot{q},t)\mathrm{d}t\\ & =& {\int }_{t_1}^{t_2}\{L(q+\delta q,\dot{q}+\delta \dot{q},t)-L(q,\dot{q},t)\}\mathrm{d}t\\ & =& {\int }_{t_1}^{t_2}[L(q,\dot{q},t)+\{\frac{\partial L(q,\dot{q},t)}{\partial q}\delta q+\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}}\delta \dot{q}\}-L(q,\dot{q},t)]\mathrm{d}t\\ & & \dots L(q+\delta q,\dot{q}+\delta \dot{q},t)\fallingdotseq L(q,\dot{q},t)+\{\frac{\partial L(q,\dot{q},t)}{\partial q}\delta q+\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}}\delta \dot{q}\}2\mathrm{次}\mathrm{以}\mathrm{降}\mathrm{を}\mathrm{無}\mathrm{視}\\ & =& {\int }_{t_1}^{t_2}\{\frac{\partial L(q,\dot{q},t)}{\partial q}\delta q+\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}}\delta \dot{q}\}\mathrm{d}t\\ & =& {\int }_{t_1}^{t_2}\frac{\partial L}{\partial q}\delta q\mathrm{d}t+{\int }_{t_1}^{t_2}\frac{\partial L}{\partial \dot{q}}\delta \dot{q}\mathrm{d}t\\ & =& {\int }_{t_1}^{t_2}\frac{\partial L}{\partial q}\delta q\mathrm{d}t+{\int }_{t_1}^{t_2}\frac{\partial L}{\partial \dot{q}}\frac{\mathrm{d}}{\mathrm{d}t}\delta q\mathrm{d}t\\ & & \dots \delta \dot{q}=\frac{\mathrm{d}}{\mathrm{d}t}\delta q\\ & =& {\int }_{t_1}^{t_2}\frac{\partial L}{\partial q}\delta q\mathrm{d}t+{\left[\frac{\partial L}{\partial \dot{q}}\delta q\right]}_{t_1}^{t_2}-{\int }_{t_1}^{t_2}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\delta q\mathrm{d}t\\ & & \dots {\int }_a^{b}fg\mathrm{d}x={\left[fG\right]}_a^b-{\int }_a^b{f}^{\prime }G\mathrm{d}x\\ & =& {\int }_{t_1}^{t_2}\frac{\partial L}{\partial q}\delta q\mathrm{d}t-{\int }_{t_1}^{t_2}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\delta q\mathrm{d}t\\ & & \dots {\left[\frac{\partial L}{\partial \dot{q}}\delta q\right]}_{t_1}^{t_2}=0\dots \delta q(t_1)=\delta q(t_2)=0\\ & =& {\int }_{t_1}^{t_2}\{\frac{\partial L}{\partial q}\delta q-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\delta q\}\mathrm{d}t\\ & =& {\int }_{t_1}^{t_2}\{\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}\}\delta q\mathrm{d}t\end{array}$$ 任意の$\delta q$において$\delta S=0$となるには$\{\}$の中が常に0となる必要がある． $$\begin{array}{rcl}\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}& =& 0\\ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}& =& 0\end{array}$$ ラグランジアンに2階以上の$q$の微分が含まれないのはつまり，位置と速度でその後の運動が決まること合致する．

積の積分 / 部分積分

積の積分 / 部分積分

$f$,$g$が$x$の凾数とし，$f^{\prime }$,$g^{\prime }$をそれぞれのxでの微分とする． $$\begin{array}{rcl}(fg{)}^{\prime }& =& f^{\prime }g+f{g}^{\prime }\dots (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }\\ f^{\prime }g& =& (fg{)}^{\prime }-f{g}^{\prime }\\ \int f^{\prime }g\mathrm{d}x& =& \int (fg{)}^{\prime }\mathrm{d}x-\int f{g}^{\prime }\mathrm{d}x\\ & =& fg-\int f{g}^{\prime }\mathrm{d}x\end{array}$$ $F$の微分を$f$として，$f$を$F$に，$f^{\prime }$を$f$に置き換える． $$\begin{array}{rcl}\int fg\mathrm{d}x& =& Fg-\int F{g}^{\prime }\mathrm{d}x\\ {\int }_a^{b}fg\mathrm{d}x& =& {\left[Fg\right]}_a^b-{\int }_a^{b}F{g}^{\prime }\mathrm{d}x\end{array}$$

積の微分

積の微分

$$\begin{array}{rcl}\frac{\mathrm{d}\{f(x)g(x)\}}{\mathrm{d}t}& =& {\displaystyle \underset{h\to 0}{lim}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}}\\ & =& \underset{h\to 0}{lim}\frac{f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h}\dots -f(x+h)g(x)+f(x+h)g(x)=0\\ & =& \underset{h\to 0}{lim}\{\frac{g(x+h)-g(x)}{h}f(x+h)+\frac{f(x+h)-f(x)}{h}g(x)\}\\ & =& \underset{h\to 0}{lim}\frac{g(x+h)-g(x)}{h}f(x+h)+\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}g(x)\\ & =& \frac{\mathrm{d}\{g(x)\}}{\mathrm{d}t}f(x)+\frac{\mathrm{d}\{f(x)\}}{\mathrm{d}t}g(x)\end{array}$$ $$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$$