Lie groups actions and curvilinear coordinates question

[mnb96]

Hello,
let's suppose I have the following system of curvilinear coordinates in ℝ²: $$x(u,v) = u$$ $$y(u,v) = v + e^u$$ where one arbitrary coordinate line $C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2}$ represents the orbit of some point in ℝ² under the action of a Lie group.

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

My questions are:

1) How can I find a system of curvilinear coordinates x(u,t), y(u,t) such that when keeping t fixed I would obtain the orbits C(u), and when keeping u fixed I would obtain the orbits K(t) ?

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 $C(u)=u \mathbf{e_1} + (u^2+\lambda) \mathbf{e_2}$, 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 orbits C(u) of the first Lie group

 

[jvicens]

Hello,

I would suggest using the method of tensor calculus to find the desired system of curvilinear coordinates. This involves finding the transformation matrix that relates the original coordinates (u,v) to the new coordinates (u,t) and (v,t). This transformation matrix can be found by solving a system of equations using the given expressions for x(u,v) and y(u,v).

As for the conditions that the two Lie groups should satisfy, they should have the same action on the orbits C(u) and K(t). This means that the transformation matrix should be the same for both groups. In other words, the two Lie groups should have the same generators, or equivalently, the same Lie algebra.

If the two Lie groups do not have the same generators, then it is not possible to find a system of curvilinear coordinates that satisfies the given conditions. In this case, the orbits C(u) and K(t) will not intersect at two points, but rather at different points, making it impossible to find a single coordinate system that describes both sets of orbits.

I hope this helps. Let me know if you have any further questions.