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At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):

$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial q_j}= \sum_i [\frac {d}{dt}(m_i {\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial \dot q_j})-m_i {\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial q_j}]$$

and

$$\sum_j \{ \frac{d}{dt}[ \frac{\partial}{\partial \dot q_j}(\sum_i \frac{1}{2}m_i v^2_i)] - \frac{\partial}{\partial q_j}(\sum_i \frac{1}{2}m_i v^2_i)-Q_j \}\delta q_j .$$

Why does "$ \frac {1}{2}$" appear in the second formula?

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  • $\begingroup$ Another question about the same equation in Goldstein: physics.stackexchange.com/q/12361/2451 $\endgroup$ – Qmechanic Oct 28 '12 at 20:58
  • $\begingroup$ @Qmechanic thank you. I was sure that I have already read a similar question but I couldn't find it! :) $\endgroup$ – sunrise Oct 29 '12 at 11:32
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The $\frac{1}{2}$ is due to the differentiation rule

$$\frac{\partial }{\partial \dot q_j}({\bf v}_i \cdot {\bf v}_i ) ~=~2{\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial \dot q_j},$$

and

$$\frac{\partial }{\partial q_j}({\bf v}_i \cdot {\bf v}_i ) ~=~2{\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial q_j}.$$

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