Are toroidal coordinates orthogonal and or separable?
From D. Goncz@21:1/5 to All on Mon Jan 29 03:11:13 2024
Well hello there readers of the classic moderated physics news group sci.physicsresearch
Many of you are aware that orthogonal coordinates are a refinement of curvilinear coordinates and the orthogonal coordinates are useful when
initial value and boundary value specifications make a problem insoluble in Cartesian coordinates
So my question is blunt.
Are toroidal coordinates orthogonal coordinates I think they are...
A toroidal coordinate system given certain limitations on the ratio of
major to minor radius is there for an orthogonal coordinate system
By inseparable I mean to refer to the laplacian
You see I didn't quite get to that in math
Just got up to div grad curl in 3d.
Now here's the fun part!
Most of the orthogonal coordinate systems are available as the
non-degenerate solutions to the generalized elliptical coordinate system because of the ordering of the parameters in a string of less than signs
well explored by author John copless excuse me that's Ms transcribed I'll
spell the name
G i a n k o p l i s.
However the topological relationship between the generating circles
defining a toroidal surface leads me to ask
Are any of the orthogonal coordinate systems including Cartesian derivable
from topological relationships having to do with sweeping perhaps, and the Cartesian product, more than likely?
It seems interesting that there might be a way to drive all conceivable
systems of three-dimensional orthogonal coordinates from the same root principles instead of what we have now which is kind of a patch together hodgepodge of symbolic math reducing everything to Cartesian and then
exploding it outwards into some new system yuck yuck yuck yuck yuck
I'm just saying I think we can do better than that