For our Classical Mechanics class, I'm reading Chapter 1 of Goldstein, et al. Now I come across Eq. (1.50). To put it in context:
$$\begin{align*} \sum_i{\dot{\mathbf{p}_i} \cdot \delta\mathbf{r}_i}&=\sum_i{m_i\ddot{\mathbf{r}}_i \cdot \delta{\mathbf{r}_i}}\\ &=\sum_{i,j}{m_i\ddot{\mathbf{r}}_i} \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j \end{align*}$$
Consider now the relation Eq. (1.50): $$\begin{align*} \sum_{i,j}{m_i\ddot{\mathbf{r}}_i} \cdot \frac{\partial\mathbf{r}_i}{\partial q_j}&= \sum_i{\left[ \frac{d}{dt} \left( m_i\dot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \right) - m_i\dot{\mathbf{r}}_i \frac{d}{dt} \left( \frac{\partial \mathbf{r}_i}{\partial q_j} \right) \right]} \end{align*}$$
I'm at a loss for how he resolved it that way. He goes on to explain that we can interchange the differentiation with respect to $t$ and $q_j$. My question is: Why is there a subtraction in Eq. (1.50)?
Answer
Why is there a subtraction in Eq. (1.50)?
Goldstein is using the Leibniz rule for differentiation of a product
$$ \frac{d (fg)}{dt}~=~\frac{d f}{dt}g + f\frac{d g}{dt} $$
with
$$f=m_i\dot{\mathbf{r}}_i $$
and
$$g=\frac{\partial \mathbf{r}_i}{\partial q_j}. $$
The minus is caused by moving a term to the other side of the equation.
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