Saturday, 13 October 2018

homework and exercises - D'Alembert's Principle: Where does $-Q_j$ come from?


This is a follow-up question to D'Alembert's Principle and the term containing the reversed effective force.


From the second term of Eq. (1.45) $$\begin{align*} \sum_i{\dot{\mathbf{p}}_i \cdot \delta{\mathbf{r}_i}} &= \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j}. \end{align*}$$


Starting from Eq. (1.50), I was able to follow that $$\begin{align*} \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j}} &= \sum_i{\left[ \frac{d}{dt} \left( m_i\mathbf{v}_i \cdot \frac{\partial\mathbf{v}_i}{\partial\dot{q}_j} \right) - m_i\mathbf{v}_i \cdot \frac{\partial\mathbf{v}_i}{\partial q_j} \right]}. \end{align*}$$



Goldstein substituted the above equation to (1.45) by saying:



... and the second term on the left-hand side of Eq. (1.45) can be expanded to



By "second term", I understood it to be the very first equation I mentioned above. Therefore this is how I understood it:


$$\begin{align*} \sum_i{\dot{\mathbf{p}}_i \cdot \delta{\mathbf{r}_i}} &= \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j}\\ &= \sum_i\left[ \frac{d}{dt}\left[ \frac{\partial}{\partial\dot{q}_j} \left( \frac{1}{2}\sum_i{m_iv_i^2} \right) \right] - \frac{\partial}{\partial q_j} \left( \frac{1}{2}\sum_i{m_iv_i^2} \right) - Q_j \right]\delta q_j\\ &= \sum_i{\left[ \frac{d}{dt} \left( \frac{\partial T}{\partial\dot{q}_j} \right) - \frac{\partial T}{\partial q_j} -Q_j \right] \delta q_j} \end{align*}$$


I was able to follow $$\begin{align*} T &= \sum_i{m_i\mathbf{v}_i} \cdot \partial\mathbf{v}_i\\ T &=\frac{1}{2} \sum_i{m_iv_i^2} \end{align*}$$


But I am at a loss: Where does $-Q_j$ come from?



Answer



Similar to Newton's 2nd law, the D'Alembert's principle has both a dynamical and a kinetic term,



$$ \sum_i (\mathbf{F}^{(a)}_i - \dot{\mathbf{p}}_i) \cdot \delta \mathbf{r}_i~=~0. \tag{1.45} $$


On one hand, the dynamical term


$$\sum_i \mathbf{F}^{(a)}_i \cdot \delta \mathbf{r}_i = \sum_j Q_j \delta q_j \tag{1.48}$$


contains the generalized force


$$Q_j=\sum_i\mathbf{F}^{(a)}_i\cdot \frac{\partial \mathbf{r}_i}{\partial q_j}.\tag{1.49}$$


On the other hand, the kinetic term


$$\dot{\mathbf{p}}_i \cdot \delta \mathbf{r}_i~=~ \sum_j{\left[ \frac{d}{dt} \left( \frac{\partial T}{\partial\dot{q}_j} \right) - \frac{\partial T}{\partial q_j} \right] \delta q_j} $$


contains the kinetic energy $T =\frac{1}{2} \sum_i{m_iv_i^2}$.


Edit: It is true that the third edition of Goldstein wrongly says




[...] and the second term on the left-hand side of Eq. (1.45) can be expanded into $$ \sum_i\left[ \frac{d}{dt}\left[ \frac{\partial}{\partial\dot{q}_j} \left( \sum_i{\frac{1}{2}m_iv_i^2} \right) \right] - \frac{\partial}{\partial q_j} \left( \sum_i{\frac{1}{2}m_iv_i^2} \right) - Q_j \right]\delta q_j. \tag{1.51b} $$



It should have read



[...] and minus the left-hand side of Eq. (1.45) can be expanded into $$ \sum_i\left[ \frac{d}{dt}\left[ \frac{\partial}{\partial\dot{q}_j} \left( \sum_i{\frac{1}{2}m_iv_i^2} \right) \right] - \frac{\partial}{\partial q_j} \left( \sum_i{\frac{1}{2}m_iv_i^2} \right) - Q_j \right]\delta q_j. \tag{1.51b} $$



The second edition does not have the $-Q_j$ term, so an unfortunate mistake was introduced during the update to the third edition. This is not the first time that I have noticed that the second edition is often more carefully written than the third edition in what concerns old material. (The third edition contains a new chapter 11 about classical chaos.)


References:



  1. H. Goldstein, Classical Mechanics; Chapter 1.



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