45403 虛功原理及歐拉-拉格朗日方程式

### 一、虛功原理

\begin{align*}\ddot X=&\,(-l\cos\theta ,-l\sin\theta ) \dot\theta^2+(-l\sin\theta ,l\cos\theta ) \ddot\theta;\\ =&\,A_0+A_1; \end{align*}

\begin{align*} F_{a_1}+F_{a_0}+f=\,&mA_1+mA_0,\\ \hbox{或 } F_a+f=\,&m\ddot X,\\ \hbox{或 } f=m\ddot X\,&-F_a; \end{align*}

\begin{align} (m\ddot X-F_a,\delta X)=0; \label{1}% (1) \end{align}

### 二、 E-L方程式和虛功原理等價

\begin{align*} \hbox{動能}\ T=\,&\frac 12 m(\dot X,\dot X )=\frac 12 m(X_u \dot u+X_v \dot v,X_u \dot u+X_v \dot v )\\ =\,&\frac 12 m[(X_u,X_u ) \dot u^2+2(X_u,X_v ) \dot u\dot v+(X_v,X_v ) \dot v^2 ], \end{align*}

\begin{align} \dfrac{d}{dt} \Big(\frac{\partial L}{\partial \dot u}\Big)-\frac{\partial L}{\partial u}=0 , \label{2}\\ % (2) \dfrac{d}{dt} \Big(\frac{\partial L}{\partial \dot v}\Big)-\frac{\partial L}{\partial v}=0. \label{3}% (3) \end{align}

\begin{align*} (m\ddot X-F_a,X_u )=\,&0, \tag*{$(2)'$}\\% (2)' (m\ddot X-F_a,X_v )=\,&0. \tag*{$(3)'$} % (3)' \end{align*}

\begin{align*} \dfrac{d}{dt} \Big(\frac{\partial T}{\partial \dot u}\Big)-\dfrac{d}{dt} \Big(\frac{\partial V}{\partial \dot v}\Big)-\frac{\partial T}{\partial u}+\frac{\partial V}{\partial v}=0 \end{align*}

\begin{align*} \frac{\partial V}{\partial u}=\,&\frac{\partial V}{\partial x} \frac{\partial x}{\partial u}+\frac{\partial V}{\partial y}\frac{\partial y}{\partial u}+\frac{\partial V}{\partial z}\frac{\partial z}{\partial u}\\ =\,&\nabla V\cdot (x_u,y_u,z_u )\\ =\,&\nabla V\cdot X_u\\ =\,&-F_a\cdot X_u, \end{align*}

\begin{align*} \dfrac{d}{dt} \Big(\frac{\partial T}{\partial \dot u}\Big)-\frac{\partial T}{\partial u}-(F_a\cdot X_u)=0. \end{align*}

\begin{align} \dfrac{d}{dt} \Big(\frac{\partial T}{\partial \dot u}\Big)-\frac{\partial T}{\partial u}=(m\ddot X\cdot X_u ). \label{4}% (4) \end{align}

\begin{align} \frac{\partial T}{\partial \dot u} =\,&m\Big(\frac{\partial\dot X}{\partial\dot u},\dot X \Big)=m(X_u,\dot X ), \label{5}\\ % (5) \hbox{及 } \frac{\partial T}{\partial u}=\,&m(X_{uu} \dot u+X_{vu} \dot v,\dot X ). \label{6}% (6) \end{align}

\begin{align*} &\hskip -20pt \dfrac{d}{dt} \big(m(X_u,\dot X )\big)-m(X_{uu} \dot u+X_{vu} \dot v,\dot X )\\ =\,&m\Big(\dfrac{d}{dt} X_u,\dot X \Big)+m(X_u,\ddot X )-m(X_{uu} \dot u+X_{vu} \dot v,\dot X )\\ =\,&m(X_{uu} \dot u+X_{uv} \dot v,\dot X )+m(X_u,\ddot X )-m(X_{uu} \dot u+X_{vu} \dot v,\dot X )\\ =\,&m(X_u,\ddot X )\\ =\,& \hbox{\eqref{4} 式右邊。} \end{align*}

### 三、 E-L方程之於單擺及簡評

E-L方程：

\begin{align*} \dfrac{d}{dt} \Big(\frac{\partial L}{\partial\dot\theta}\Big)-\frac{\partial L}{\partial\theta} =\,&0,\\ \dfrac{d}{dt} (ml^2 \dot\theta )+mg\,l\sin\theta =\,&0,\\ \hbox{或} ml^2 \ddot\theta +mg\,l\sin\theta =\,&0. \end{align*}

\begin{align*} \dfrac{d}{dt} \Big(\frac{\partial L}{\partial \dot u}\Big)-\frac{\partial L}{\partial u}=0 , \\ \dfrac{d}{dt} \Big(\frac{\partial L}{\partial \dot v}\Big)-\frac{\partial L}{\partial v}=0. \end{align*}

### 參考資料

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics; 60, 1978. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., 1976. H. Goldstein, Classical Mechanics, Addison-Wesley series in Advanced Physics, 1962.

---本文作者為台大數學系退休教授---