Symplectic Reduction & Coadjoint Orbits of SO(3)

Symplectic reduction (Marsden-Weinstein 1974) shows that conserved quantities correspond to symmetries via the momentum map J: T*Q → g*. For SO(3), coadjoint orbits are spheres in ℝ³ — the geometric home of angular momentum. Each sphere is itself a symplectic manifold.

Initial L_x: 1.00

Initial L_y: 2.00

Initial L_z: 0.50

Inertia ratio I₁/I₂: 1.50

View

|L|² = — (conserved)
L_z = —
Energy = —
Orbit type: —

L = (L_x, L_y, L_z) ∈ so(3)* coadjoint orbit: |L|² = const Euler's eqns: İ = L × (I⁻¹L)