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:
∑i˙pi⋅δri=∑imi¨ri⋅δri=∑i,jmi¨ri⋅∂ri∂qjδqj
Consider now the relation Eq. (1.50): ∑i,jmi¨ri⋅∂ri∂qj=∑i[ddt(mi˙ri⋅∂ri∂qj)−mi˙riddt(∂ri∂qj)]
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 qj. 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
d(fg)dt = dfdtg+fdgdt
with
f=mi˙ri
and
g=∂ri∂qj.
The minus is caused by moving a term to the other side of the equation.
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