Old papers that classify separable coordinate systems

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Old papers that classify separable coordinate systems

Postby PWrong » Thu Jun 11, 2015 2:33 am

I asked about counting separable coordinate systems in higher dimensions on Mathoverflow. I didn't want to give anything away so I focused on the PDEs rather than mentioning toratopes or the forum. I got two excellent answers which you can read here:
http://mathoverflow.net/questions/208901/separable-coordinate-systems-for-the-laplace-and-helmholtz-equations

What I want to show you in particular are these papers:

The papers are fairly intractable, and a lot of the symmetry notation is unfamiliar to me. But in both papers they are talking about coordinate systems based on rotatopes. Specifically, they have a list of coordinate systems that includes all the rotatope-based coordinate systems, but also includes parabolic and elliptic-type coordinates. They categorise them based on partitions of n, just like us. Personally I think the notation they use is horrible though. I haven't had time to read them thoroughly, but I think they also talk about cyclidal coordinate systems, which correspond to the quartic toratopes.
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Re: Old papers that classify separable coordinate systems

Postby ICN5D » Thu Jun 11, 2015 5:14 am

Well, that's an interesting find. I think I read somewhere how part of math is the invention of better notations! Plus, people are mentioning some potential real-world applications to the PDE's and coordinate systems in engineering. Not sure how related it is, but I read somewhere else how robotic CNC arms compute their positions by solving the quartic equation of a torus. A more complex armature would need a higher dimensional surface as a reference, which makes me think of higher toratopes.
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Re: Old papers that classify separable coordinate systems

Postby PWrong » Fri Jun 12, 2015 2:59 am

Certainly an interesting find, although the answers are a bit discouraging. The main issue I see is that exact solutions to PDEs are less interesting now that we have computers. Higher dimensional coordinate systems may still be useful, but the applications will likely require maths that I haven't learned yet.

I'm thinking I could still publish in a less serious (but still good) journal, intended for a wider audience. At the moment I'm working on another project, which could take anywhere from a few months to a few years to finish.
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