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let's suppose I have the following system of curvilinear coordinates in ℝ^{2}: [tex]x(u,v) = u[/tex] [tex]y(u,v) = v + e^u[/tex] where one arbitrarycoordinate line[itex]C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2}[/itex] represents theorbitof some point in ℝ^{2}under the action of aLie group.

Now I consider another Lie group that produces the same orbits, but rotated by 90°, that is: [tex]K_s(t)=(e^{-t}+s) \mathbf{e_1} + t \mathbf{e_2}[/tex].

My questions are:

1) How can I find a system of curvilinear coordinatesx(u,t),y(u,t)such that when keepingtfixed I would obtain the orbitsCfixed I would obtain the orbits K(t) ?(u), and when keepingu

2) What are the conditions that these two Lie groups should satisfy in order for this to be possible? For example, had our orbits been paraboloids of the kind [itex]C(u)=u \mathbf{e_1} + (u^2+\lambda) \mathbf{e_2}[/itex], then no such coordinate system can exist because the other Lie group that produces the parabolae rotated by 90° will give orbits that intersects at two points with the orbitsC(u)of the first Lie group

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# Lie groups actions and curvilinear coordinates question

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