Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 355chart is(ϕ, ψ, θ) ∈ R 3 ↦−→ A ∈ SO(3), 0 < ϕ, ψ < 2π; 0 < θ < π, where⎡⎤cos ψ cos ϕ − cos θ sin ϕ sin ψ cos ψ cos ϕ + cos θ cos ϕ sin ψ sin θ sin ψA = ⎣ − sin ψ cos ϕ − cos θ sin ϕ sin ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ψ ⎦sin θ sin ϕ − sin θ cos ϕ cos θThe corresponding conjugate momenta are denoted by p ϕ , p ψ , p θ , so{ϕ, ψ, θ, p ϕ , p ψ , p θ } is the phase–space T ∗ SO(3). Thus, we haveM = T ∗ SO(3), ω = dp ϕ ∧ dϕ + dp ψ ∧ dψ + dp θ ∧ dθ, H = 1 2 K,K = [(p ϕ − p ψ cos θ) sin ψ + p θ sin θ cos ψ] 2I 1 sin 2 θ+ [(p ϕ − p ψ cos θ) cos ψ − p θ sin θ sin ψ] 2I 2 sin 2 + p2 ψ,θI 3where I 1 , I 2 , I 3 are the moments of inertia, diagonalizing the inertia tensorof the body.The Hamiltonian equations are˙ϕ = ∂H∂p ϕ,ṗ ϕ = − ∂H∂ϕ ,˙ψ =∂H∂p ψ,ṗ ψ = − ∂H∂ψ ,˙θ =∂H∂p θ,ṗ θ = − ∂H∂θ .For each f, g ∈ C ∞ (T ∗ SO(3), R) the Poisson bracket is given by{f, g} ω = ∂f ∂g− ∂f ∂g∂ϕ ∂p ϕ ∂p ϕ ∂ϕ + ∂f ∂g− ∂f ∂g∂ψ ∂p ψ ∂p ψ ∂ψ+ ∂f ∂g− ∂f ∂g∂θ ∂p θ ∂p θ ∂θ .The Heavy Top – ContinuedRecall (see (3.8.4.2) above) that the heavy top is by definition a rigidbody moving about a fixed point in a 3D space [Puta (1993)]. The rigidityof the top means that the distances between points of the body are fixed