# Symmetry, Transformations and non-linear transformations

I am a physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) or differences 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, if not what are the cases in which we can't associate such generators to transformation.

I have already posted this question here. However I have not obtained satisfactory responses there.

• For any student of physics it should never be too soon to understand that coordinates are utterly irrelevant in physics (defining physical quantities; measuring their values; forming expectations about values still to be obtained). Concrete example: given a metric space $(S, d)$, with $d : S \times S \rightarrow \mathbb R$ , as measured values, they remain unaltered under any (subsequent) assignment of coordinates $x : S \rightarrow \mathbb R^n$, together with the thereby induced function $g : \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R$, such that $g(x(A), x(B)) = d( A, B )$. – user12262 Jan 18 '14 at 11:16
• @user12262 : Thanks a lot, I was expecting something like this !! However, can you elucidate this with some example which is non-trivial (say for instance the generator associated to the transform for Cartesian to Parabolic coordinates). – user35952 Jan 19 '14 at 3:29
• user35952: "can you elucidate this with some example [...] the generator associated to [...]" -- Well, in order to elucidate this in (less than) 500 characters available, I'd suggest to submit the question "How do you call the operator $-i \frac{d}{d \kappa}$ (if not plainly: the generator of curvature)?" under the tag terminology. Also, as a student of physics, I'd refer to the definition of "curvature $\kappa$" indicated here: orbit.dtu.dk/en/publications/… – user12262 Jan 19 '14 at 9:10
• p.s. Correction: the number a characters admissible for writing a comment here is not limited to 500 but to 600. (Which I nevertheless had used up in the preceding comment. If the limit had instead been set as generously as even 606 characters, at least, then I might have suggested to ask more correctly about naming the operator "$-i \hbar \frac{d}{d \kappa}$".) – user12262 Jan 20 '14 at 6:02

## 1 Answer

My immediate response would be the same as you had at algebra forums. In what way were those answers not satisfactory?

Do you know what each of the transformations do?

Can you describe them in words?

Taking comments on board: Examples of "describing them in words" that may help you think about the question:

A coordinate transform shifts the axes around, a coordinate-system transform changes the axes themselves.

The first keeps the basis vectors the same type - the second changes to an entirely different kind of basis.

• I did the math for the non-linear transforms and I could see they don't preserve the form of equations describing physical systems. However I was not able to associate this to some group theory to see if the existence of generators can be proved (or disproved) for such non-linear transformations. – user35952 Jan 17 '14 at 4:15
• @Simon Brodge: you should have posted this as a comment. – user28355 Jan 17 '14 at 4:27
• @user35952 OP: What education level is the question being answered in? Note: are the transforms above linear or non-linear? – Simon Bridge Jan 17 '14 at 4:48
• @SimonBridge : I am sorry, my comment should have gone like, "Although am still NOT able to see where idea of generators come up in the case of non-linear transformations (if at all they do)." With my knowledge, I am thinking rotations and other symmetry transformations are linear (at least I know only linear symmetry transformations).I believe the second kind of transformations I have mentioned in the question is non-linear. – user35952 Jan 17 '14 at 5:24
• Still need to know where to pitch the answer. Off last comment: linear fersure - so the fact they are both linear is a set of similarities right? – Simon Bridge Jan 18 '14 at 0:11