# 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 $g(x(A), x(B)) = d( A, B )$. Jan 18, 2014 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). Jan 19, 2014 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/… Jan 19, 2014 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 "$-i \hbar \frac{d}{d \kappa}$".) Jan 20, 2014 at 6:02

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?