Assume \(\bm F\) is active force and \(\bm R\) is constraint force. Then for \(i\)th body we have Newton’s Second Law:

$$\bm F^\mathrm{tot}_i=\bm F_i+\bm R_i=m\bm a_i$$

With a virtual displacement \(\delta \bm x_i\), we define:

$$\delta W_i=\bm F^\mathrm{tot}_i\cdot\delta\bm x_i,\quad \delta K_i=m\bm a_i\cdot\delta\bm x_i$$

Then we have \(\delta W_i=\delta K_i\). Define

$$\delta W=\sum_i\delta W_i,\quad\delta K=\sum_i\delta K_i$$

and obviously

$$\delta W=\delta K$$

As the constraints will not do work, we can rewrite

$$\delta W=\sum_i\bm F_i\cdot\delta\bm x_i$$


Statics \(\delta K=0\)

Principle of Virtual Work:

$$\frac{\delta W}{\delta q}=0$$

i.e. Minimal Potential Energy

$$Q\mdef-\frac{\delta V}{\delta q}=0$$


d’Alembert’s Principle—Counterpart of Principle of Virtual Work:

$$\frac{\delta W}{\delta q}=\frac{\delta K}{\delta q}$$


$$\begin{aligned} Q\mdef\bm F_i\cdot\frac{\pp \bm x_i}{\pp q}&=m_i\bm a_i\cdot\frac{\pp \bm x_i}{\pp q}\\ &=m_i\frac{\dd \bm v_i}{\dd t}\cdot\frac{\pp \bm x_i}{\pp q}\\ &=m_i\frac{\dd}{\dd t}\left( \bm v\cdot\frac{\pp \bm x_i}{\pp q}\right)-m_i\bm v\cdot\frac{\dd}{\dd t}\frac{\pp}{\pp q}\bm x_i\\ &=m_i\frac{\dd}{\dd t}\left(\bm v\cdot\frac{\pp \dot{\bm x_i}}{\pp\dot q}\right)-m_i\bm v\cdot\frac{\pp}{\pp q}\frac{\dd}{\dd t}\bm x_i\\ &=\frac{\dd}{\dd t}\left(m_i\bm v_i\cdot\frac{\pp\bm v_i}{\pp\dot q}\right)-m_i\bm v_i\cdot\frac{\pp \bm v_i}{\pp q}\\ &=\frac{\dd}{\dd t}\frac{\pp T}{\pp\dot q}-\frac{\pp T}{\pp q}\\ &=\left(\frac{\dd}{\dd t}\frac{\pp}{\pp\dot q}-\frac{\pp}{\pp q}\right)T\label{T}\end{aligned}$$

For conservative or monogenic system,

$$Q=\left(\frac{\dd}{\dd t}\frac{\pp}{\pp\dot q}-\frac{\pp}{\pp q}\right)V\label{V}$$

As a result of last two equations, if we define \(L=T-V\), we have

$$\left(\frac{\dd}{\dd t}\frac{\pp}{\pp\dot q}-\frac{\pp}{\pp q}\right)L=0$$


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