Thus the Hamiltonian is H = P θ θ θ θ ˙ ˙ + P φ φ φ φ ˙ ˙ − − L where P θ θ = ∂ ∂ L ∂ ∂ θ θ ˙ ˙ = m ℓ ℓ 2 θ θ ˙ ˙ and P φ φ = ∂ ∂ L ∂ ∂ φ φ ˙ ˙ = m ℓ ℓ 2 sin 2 θ θ φ φ ˙ ˙ . In terms of coordinates and momenta, the Hamiltonian reads H = [ 1 2 m ℓ ℓ 2 θ θ ˙ ˙ 2 + 1 2 m ℓ ℓ 2 sin 2 θ θ φ φ ˙ ˙ 2 ] ⏟ ⏟ T + [ − − m g ℓ ℓ cos ⁡ ⁡ θ θ ] ⏟ ⏟ V = P θ θ 2 2 m ℓ ℓ 2 + P φ φ 2 2 m ℓ ℓ 2 sin 2 ⁡ ⁡ θ θ − − m g ℓ ℓ cos ⁡ ⁡ θ θ . Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations, θ θ ˙ ˙ = P θ θ m ℓ ℓ 2 φ φ ˙ ˙ = P φ φ m ℓ ℓ 2 sin 2 ⁡ ⁡ θ θ P θ θ ˙ ˙ = P φ φ 2 m ℓ ℓ 2 sin 3 ⁡ ⁡ θ θ cos ⁡ ⁡ θ θ − − m g ℓ ℓ sin ⁡ ⁡ θ θ P φ φ ˙ ˙ = 0. Momentum ⁠ P φ φ ⁠, which corresponds to the vertical component of angular momentum ⁠ L z = ℓ ℓ sin ⁡ ⁡ θ θ × × m ℓ ℓ sin ⁡ ⁡ θ θ φ φ ˙ ˙ ⁠, is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis. Being absent from the Hamiltonian, azimuth φ φ is a cyclic coordinate, which implies conservation of its conjugate momentum.