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Re: Example of Non-Orthogonal Coordinates

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On 6/28/18 8:37 AM, F. Russell wrote:
> Orthogonal coordinate systems are the most common
> because they are easy to use.  Wikipedia has a list
> of orthogonal coordinate systems:
> 
> https://en.wikipedia.org/wiki/Orthogonal_coordinates
> 
> I need some examples of non-orthogonal coordinate systems.
> The reason is that ALL textbooks present non-orthogonal
> systems in a totally general way and I want to practice on some
> actual examples rather than dry generalized systems.
> 
> Can anyone provide an example or examples of non-orthogonal
> coordinate systems?  By an example I mean the actual functions
> that relate the Cartesian variables (x, y, z) to the
> non-orthogonal coordinates (q1, q2, q3): x=f1(q1, q2, q3),
> y=f2(q1, q2, q3), z=f3(q1, q2, q3) or the inverse relations.

Almost all commonly used spatial coordinate systems are
orthogonal, simply because they're usually easier to work
with.  One exception is the case of coordinates in a crystal
lattice, as another poster mentioned, but these are just
skewed.

If you're willing to add time as a coordinate, the most common
nonorthogonal coordinate systems are those that describe a
rotating body.  Start with spherical or cylindrical coordinates
and make a shift \phi -> \phi - \omega t, where omega is the
angular velocity.  A standard exercise is to then work out
the Coriolis effect in the new coordinates.

There are other variations of this idea -- for instance, the
rest frame coordinates for an object moving along an arbitrary
trajectory.  But for spatial coordinates alone, I can't think
of any commonly used examples more complicated than skew.

Steve Carlip

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Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 15:37 +0000
  Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 09:45 -0700
    Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 16:57 +0000
      Re: Example of Non-Orthogonal Coordinates konyberg <konyberg@online.no> - 2018-06-28 10:27 -0700
        Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 17:41 +0000
          Re: Example of Non-Orthogonal Coordinates konyberg <konyberg@online.no> - 2018-06-28 11:14 -0700
        Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 10:42 -0700
          Re: Example of Non-Orthogonal Coordinates bursejan@gmail.com - 2018-06-28 10:49 -0700
            Re: Example of Non-Orthogonal Coordinates "F. Russell" <fr@random.info> - 2018-06-28 18:10 +0000
            Re: Example of Non-Orthogonal Coordinates bursejan@gmail.com - 2018-06-28 11:12 -0700
              Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 12:09 -0700
                Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 14:05 -0700
            Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 11:14 -0700
      Re: Example of Non-Orthogonal Coordinates "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-28 10:28 -0700
  Re: Example of Non-Orthogonal Coordinates FredJeffries <fredjeffries@gmail.com> - 2018-06-29 06:20 -0700
  Re: Example of Non-Orthogonal Coordinates Steven Carlip <carlip@physics.ucdavis.edu> - 2018-06-29 14:06 -0700

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