Hyperbolic

Hyperbolic plane: H is given by \(z^2=x^2+y^2+1\) (with z ≥ 1) (R³ with Minkowski metric)

Or H is the subset of \(\mathfrak{sl}(2,\,\mathbb{R})\) defined by determinant = 1

Symmetry group: G = SL(2, R)
Hyperbolic plane is coadjoint orbit.

Phase space M = H x H x . . . x H (n copies)

Momentum map: J(A₁, . . . A_n) = Σ κ_j A_j.

Momentum isotropy:

then det(μ) > 0 and G_μ = SO(2).
also J^-1(μ) is bounded:
- use z-components of points: Σ_j κ_j z_j is fixed (= z(μ)). So all z_j < z(μ)/min(κ_i)

For each μ, J^-1(μ) is 4 - 3 = 1-dimensional, so (if connected) it must concide with the G_μ-orbit. In particular, this means reduced spaces are points. (But how is reduction done for non-compact group?)
μ≠0 (unless A₁=A₂ and κ₁=-κ₂)
if κ₁κ₂ > 0 then motion is periodic as G_μ = SO(2).
if κ₁κ₂ < 0 then motion cen be either bounded or unbounded (depending on μ)
- When is det(μ)=0?
Is κ₂ = -κ₁ special in any way?

Is (closure of) J^-1(μ) homeo to S³? (and reduced space is S² and so predict no of REs: 3 collisions (maxima of H) so 2 saddles and a minimum at least)

JAM -Vortex wiki