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I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds :

  1. Rotation of coordinate (and hence new transformed coordinate system) or translation and so on. (Most of which are symmetry associated)
  2. Transformation to a different system of coordinates - like cartesian to cylindrical or spherical to parabolic and so on.

In this regard I would like to know the difference between these two in the linear algebra language. I can see in both cases the inner product is preserved.

Secondly, I am also interested in knowing if we can associate symmetry to the second kind of transformation and hence some generator of transformation.

Thanks

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Rotation, translation etc are linear transformations. They map straight lines to straight lines and quadratic curves to quadratic curves.

cylindrical, spherical etc. are non linear. They do not preserve lines etc.

In Physics, you also have contranomial transformation. In mathematics, they are called adjoints. So for every transformation of position, there is a corresponding transformation of force, so that force times position (work) does not change. One important principle that must be maintained in physics (and not general mathematical transformations) are the conservation laws (energy, power, momentum etc).

Added in response to OP's comment

To keep it simple, I will work in 2-D.

Suppose $(x,y)$ is a point. Then the transformation $$ X = x+y \\ Y = 2x-y $$ is an example of linear transformation from original (lower case) to the new (upper case). You can transform back using $$ x=(X+Y)/3 \\ y=(2X-Y)/3 $$ Suppose we have a line in the original coordinate given by $$ y = 5 x + 1 $$ then in the new coordinates, the same line will be $$ (2X-Y)/3 = 5 (X+Y)/3+1 \Rightarrow Y = -1/2 X - 1/2$$ which is a straight line. You can do the same for quadratic curves also. The transformation is written using matrices as $$\begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & -1\end{pmatrix} \, \begin{pmatrix} x \\ y \end{pmatrix} $$ and the inverse transform as $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & -1\end{pmatrix} ^{-1} \, \begin{pmatrix} X \\ Y \end{pmatrix} $$

If the forces are $f$ and $g$ in the original co-ordinates, and $F$ and $G$ in the new coordinates, then the contranomial transform is given by $$\begin{pmatrix} f \\ g \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & -1\end{pmatrix} ^{T} \, \begin{pmatrix} F \\ G \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 1 & -1\end{pmatrix} \, \begin{pmatrix} F \\ G \end{pmatrix} $$ Note that the transform is from new to old and in the place of inverse you have a transpose. Now if $x,y$ changes by $dx$, $dy$, then in new coordinates we have $$ dX = dx + dy\\dY = 2 dx - dy $$ and the incremental work done (in new coordinates) is $$ F dX + G dY = F (dx + dy) + G (2dx -dy)\\ = (F+2G) dx + (F-G) dy = f dx + g dy $$ This shows conservation laws of mechanics will hold in both the coordinates.

For an example of non-linear transform, consider polar form $$ r = \sqrt{x^2+y^2} \\ \theta = \arctan2(x,y) $$ Equation of the line would be a messy formula in terms of $r$ and $\theta$, and I will leave it to you to see what the contranomial transforms should be (not easy!)

Hope this long explanation helps.

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  • $\begingroup$ Thanks, I can understand the difference of linearity and non-linearity. Can you explicitly explain one example in linear algebra language (elucidating both cases). Thanks again ! $\endgroup$ – user35952 Jan 5 '14 at 8:34
  • $\begingroup$ Let me edit my answer... $\endgroup$ – user44197 Jan 5 '14 at 14:47
  • $\begingroup$ Thank you so much !! The elaborate explanation was really very helpful. Although am still able to see where idea of generators come up in the case of non-linear transformations (if at all they do). If they don't why is that we don't have such things. However, thanks a ton for the explanation, was a neat work. $\endgroup$ – user35952 Jan 6 '14 at 17:49

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