Coordinate Systems After Deformation of Axes

[Opus_723]

Disclaimer: I am a physics student and I have very little knowledge of topology or differential geometry. I don't necessarily expect a complete answer to this question, but I haven't really found any reference that approaches what I'm trying to ask, so I'd be quite happy to simply be pointed in the right direction, even just advice on what terms to google.

Suppose I have a typical Cartesian xy coordinate system. Now suppose I replace the x-axis and the y-axis with curves (but just gentle curves, they still extend to infinity and do not intersect or anything crazy). Given the shapes of these new axes, is there some natural or "nice" way to define new coordinates for the entire 2D space? I realize you could probably do many things arbitrarily depending on what properties you want your coordinates to have (first thought that came to mind was to use the minimum distance to each curve), but I was wondering if people who are familiar with topology would know of any choices that are particularly intuitive or have particularly nice properties, and some literature behind them.

I'm actually interested in higher dimensional versions of this, where you deform all of the coordinate planes, but I can't even satisfy myself in the 2D case, so I thought I'd focus on that.

Sorry this is such a vague question, but I've been searching for anything related to this and I can't seem to find a foothold anywhere, except that I should learn some basic topology/differential geometry, but I'm looking for a little more direction than that so I know what to work towards.

 

[jambaugh]

What you are referring to is called curvilinear coordinates and if you search those terms you'll find a wealth of information online. One observation about your question is that you are focusing on the axes. Note that the axes alone do not define the coordinate system but the axes plus the convention of drawing perpendicular lines back to them from the point whose coordinates you wish to define. (Plus also the choice of how you parameterize the axes, see below.)

But I think that for you the best starting point is to work functionally (as in using functions). Coordinates are defined as a mapping between a sequence of numbers (coordinates) and points on the plane or in space or in a higher dimensional abstract space. The first non-Cartesian coordinate system students experience is polar coordinates so look at that as a good paradigm example.

The coordinate lines (or in general curves) are the sets of points obtained by holding all but one of the coordinates fixed. Likewise coordinate planes (or in general surfaces) are obtained by allowing only two of the coordinates to vary, and so on in higher dimensions.

Notice that you can also change coordinate systems while leaving the coordinate lines or curves unchanged. As an example consider graphs which have a logarithmic scale. You can in fact implement "curvilinear coordinates" in only one dimension by reparametrizing the number line.

You should see a reasonable introduction of curvilinear coordinates and their mathematical implications in multivariable calculus (usually your third calc. course). I have played with the idea of writing a textbook to go beyond what is usually taught and fill the gap between Calc III and full blown Differential Geometry but that is a big undertaking. Maybe with the current situation I should pick up the project again.

But I would say that if you want to understand this fully (and a physics student would be well served in doing so) start with Multivariable Calculus and Linear Algebra, then pick up an introductory differential geometry text. A bit of abstract algebra (mainly group theory) and an introduction to Lie groups are also very helpful and rounding out your understanding.

A final note. Your question implies that you are still thinking in terms of Euclidean geometry while changing the coordinate system. In differential geometry one is also changing the geometry which will remove the ability to even define a Cartesian coordinate system. It is because of this last point that general coordinates are fully explored in differential geometry but I think it is helpful to keep in mind that there are two things going on there, change of coordinates and change of geometry.

 

[Opus_723]

Thanks for responding! I'm actually pretty familiar with curvilinear coordinates, where you have families of surfaces. I should probably be more specific about what I'm interested in:

In this case the space I am starting with is actually a 2N-dimensional Hamiltonian phase space, with positions and momentum as coordinates. I also have some set of 2N "special" 2N-1-dimensional planes that partition the phase space. I know how to build a new coordinate system that uses these special planes as coordinate planes, such that on each one a particular coordinate associated with that plane is always zero, because that is just a linear transformation of coordinates. But now I have some rule that produces new "special" surfaces that can be interpreted as deformations of these planes, and I would like to build a coordinate system that uses the new deformed surfaces as "surfaces where one of the coordinates is zero" (I'm sure there must be a better name for that).

My problem is that because I don't have entire families of curves, this seems like not enough information to build a coordinate system around. I only have a set of surfaces that partition the space. And my intuition from the linear case where the special surfaces are planes is failing me, because I can't see how to generalize it. So I'm looking for guidance as to what a natural way to proceed might be. How much am I lacking? What sorts of extra information do I need to make a coordinate system that treats each surface as a surface of zero coordinate. Is there any particularly obvious choice with nice properties?

And yes, I am extremely fuzzy on whether or not my geometry is changing along with my coordinates and what that might imply.

I'll probably just have to read through a differential geometry text like you said, which is perfectly fine. I was hoping, however, to get a better idea of what to keep an eye out for as I do so.