B How to find the infinitesimal coordinate transform along a hyperbola?

AI Thread Summary
The infinitesimal coordinate transformations along a hyperbola defined by the equation b(dy)^2 - a(dx)^2 = r are expressed as δx = bwy and δy = awx, where w is related to the angle of rotation. The function w is speculated to be similar to sinh(theta), although this remains unconfirmed. The discussion seeks clarification on the derivation of these transformations and their role in preserving the invariant r. The context of this inquiry is linked to previous insights on the relationship between Lie groups and physics. Understanding these transformations is crucial for grasping the geometric properties of hyperbolas in relation to rotation.
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I've been told that the infinitesimal change in coordinates x and y takes the form δx=wy and δy=wx, and I was hoping someone could help me figure out why.
I've been told that the infinitesimal change in coordinates x and y as you rotate along a hyperbola that fits the equation b(dy)^2-a(dx)^2=r takes the form δx=bwy and δy=awx, where w is a function of the angle of rotation (I'm pretty sure it's something like sinh(theta) but it wasn't clarified for me so I'm not 100% sure). However, I'm not sure why this is the case, and I was hoping someone could show me how you get these infantesimal transformations and how they preserve the invariant r.
 
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The last time I saw this point of view was when I wrote my insight article When Lie Groups Became Physics based on a book from 1911! I have tried to use rotation as an example (the equations in brackets [...]) so maybe it helps.
 
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