We consider a plane pendulum with frequency ω ω under the effect of Earth's rotation whose angular velocity is Ω Ω → → = ( Ω Ω x , Ω Ω y , Ω Ω z ) with amplitude denoted as Ω Ω = | Ω Ω → → | . Here, the z direction points from the center of the Earth to the pendulum. The Lagrangian for the pendulum is L = 1 2 m ( x ˙ ˙ 2 + y ˙ ˙ 2 ) − − 1 2 m ω ω 2 ( x 2 + y 2 ) + m Ω Ω z ( x y ˙ ˙ − − y x ˙ ˙ ) The corresponding motion equation is x ¨ ¨ + ω ω 2 x = 2 Ω Ω z y ˙ ˙ y ¨ ¨ + ω ω 2 y = − − 2 Ω Ω z x ˙ ˙ We then introduce an auxiliary variable ϖ ϖ = x + i y that is in fact an angle variable. We now have an equation for ϖ ϖ : ϖ ϖ ¨ ¨ + ω ω 2 ϖ ϖ = − − 2 i Ω Ω z ϖ ϖ ˙ ˙ From its characteristic equation λ λ 2 + ω ω 2 = − − 2 i Ω Ω z λ λ we obtain its characteristic root (we note that Ω Ω ≪ ≪ ω ω ) λ λ = − − i Ω Ω z ± ± i Ω Ω z 2 + ω ω 2 ≈ ≈ − − i Ω Ω z ± ± i ω ω The solution is then ϖ ϖ = e − − i Ω Ω z t ( A e i ω ω t + B e − − i ω ω t ) After the Earth rotates one full rotation that is T = 2 π π / Ω Ω ≈ ≈ 24 h , we have the phase change for ϖ ϖ Δ Δ φ φ = 2 π π ω ω Ω Ω + 2 π π Ω Ω z Ω Ω The first term is due to dynamic effect of the pendulum and is termed as the dynamic phase, while the second term representing a geometric phase that is essentially the Hannay angle θ θ H = 2 π π Ω Ω z Ω Ω