Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Curvilinear coordinates from orbits

  1. Dec 15, 2011 #1
    Hello,

    I have the following problem where I have two groups of transformations [itex]R_\alpha[/itex] (rotation) and [itex]S_\lambda[/itex] (scaling) acting on the plane, so that the orbits of any arbitrary point x=(x0,y0) under the actions of [itex]S_\lambda[/itex] and [itex]R_\alpha[/itex] are known (in the former case they are straight lines from the origin; in the latter case they are concentric circles with center in 0).

    From this information, how can I "build" a system of curvilinear coordinates, where the coordinates are exactly the parameters (α,λ) of the transformations?

    PS: I know that the answer leads to the log-polar coordinates, but I need a procedure to arrive at it.
     
    Last edited: Dec 15, 2011
  2. jcsd
  3. Dec 15, 2011 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The rotation group is diffeomorphic to the circle S^1, while the scaling group is diffeomorphic to the real line. It is clear that if you take the standard coordinates θ on S^1 and r on R^1, then the map

    (-pi,pi) x (0,infty)-->R x S --->R^2
    (θ,r)------------->(Rθ,Sr)--->(Rθ[itex]\bullet[/itex]Sr[itex]\bullet[/itex](1,0)

    is a diffeomorphism onto R²\{the x<=0 ray} commonly refered to as "the polar coordinates on R^2"! ("[itex]\bullet[/itex]" stands for the group action)

    But there is no Log anywhere in this construction so perhaps this is not what you are after?
     
  4. Dec 15, 2011 #3
    Hi quasar987,

    thanks! your explanation was pretty clear. I simply didn't think of making such observations in terms of diffeomorphisms. And by the way you are right, the log-polar is not important in this case. I was just a bit confused.

    One more thing I would be very interested to know. If I replace the rotation and scaling groups with two different and not well-known groups of transformations, can we do the same? What is the procedure to find a system of curvilinear coordinates in the plane, when we know the orbits of the points under the action of the two groups?
     
  5. Dec 15, 2011 #4

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Well, in general, no, the analogue of the map we created for general group is not going to be a diffeomorphism. For instance, if instead of S^1 and R^1, you take two R^1's (both acting in the same way on R²), then we see fairly easily that the differential will be at every point non surjective. Similarly if we take two S^1's. The key for the above example to work was that the orbits are transverse.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Curvilinear coordinates from orbits
Loading...