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I'm currently working through an example question in Patrick Hamill's 'A Student's Guide to Hamiltonians and Lagrangians'. The question I'm having conceptual difficulty with is:

A particle is acted upon with components $F_x$ and $F_y$. Determine the generalized forces, $Q_i$, in polar coordinates.

It's simple to find the generalised force dependence on $r$: ($Q_r = F_x\cos(\theta) + F_y\sin(\theta)$) however I can't seem to get my head around the answer for the dependence on $\theta$, (which is $-F_x r\sin(\theta) + F_y r\cos(\theta)$).

I think that the difficulty I'm having stems from $\theta$ looking a lot like a vector here in component form. As far as I am aware, angles are never vectors.

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There is a distinction between points and vectors. Points are positions in space, and vectors are directions. One can easily mix up the two, because in Euklidean space they look rather similar.

$\theta$ in this case is a coordinate, i.e. part of the description of a point. The vector associated to that coordinate could be called $\hat{e}_\theta$, and point in the direction, in which $\theta$ changes. So, in polar coordinates, the force depends on the position at which it is evaluated.

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  • $\begingroup$ Now I get it. I was confusing the magnitude of r, with the radial vector, and the angle with the tangential vector. Thanks for pointing me in the correct direction (no pun intended). $\endgroup$ – TheJerseyChemist Jul 17 '14 at 9:54

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