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I am a Christian. Although I have an abiding interest in science and philosophy, I view things from a distinctly Christian vantage point, which harmonizes human well-being with what we know of the natural world.

Here's a nice quote I came across:

"The spectacle of the universe seems all the more grand and beautiful and worthy of its Author, when one considers that it is all derived from a small number of laws laid down most wisely." -Maupertuis, 1746


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awarded  Popular Question
Dec
8
comment What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa?
I understand that it isn't defined at the origin. However, a displacement vector $A$ that starts at the origin only has a radial component, when we consider it in polar coordinates.
Dec
8
comment What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa?
I don't understand the $A_\theta\hat{\theta}$ term. Shouldn't it vanish? After all, a vector starting from the origin only has magnitude in the radial direction, right?
Dec
7
asked What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa?
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comment Virtual Work: How is the applied force related to the coordinates chosen?
Thanks. It seems a lot clearer now. BTW, do you think Goldstein is guilty of some imprecise language here? It almost looks like he's saying that if you use the $q_i$'s you can set $\bf F_i^{(a)}$ to zero, which is obviously not the case.
Sep
12
comment Virtual Work: How is the applied force related to the coordinates chosen?
Thanks. What about the second part :"In order to equate the coefficients to zero, we must transform the principle into a form involving the virtual displacements of the $q_i$, which are independent." According to me, the vector directions of the constrained displacements won't change. They will still be given by the directions of the $\bf \delta r$'s.Then why is he saying that?
Sep
10
answered Virtual Work: How is the applied force related to the coordinates chosen?
Sep
8
comment Virtual Work: How is the applied force related to the coordinates chosen?
I have two more questions: (1) How does $\mathbf r \cdot \delta \mathbf r = 0$ imply that $x$, $y$, $z$ cannot be varied independently of one another? (2) How does the fact that $\theta$ and $\phi$ can be varied independently allow us to set ${\bf F_i^{(a)}} = 0$? Regarding the first question, does it have anything to do with it Linear Algebra?
Sep
8
comment Virtual Work: How is the applied force related to the coordinates chosen?
Thanks. However, I am still confused over Goldstein's insistence on $\it independence$ of the coordinates chosen. (see Goldstein's quote above)
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asked Virtual Work: How is the applied force related to the coordinates chosen?
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