Similarly, one can formulate the EOMs in terms of the generalized Euler angles {θ n , ϕ n }. The EOMs with these variables are a bit more complicated and non-linear in terms of {θ n , ϕ n }, as oppose to the case of {Θ n , ϕ n }. This is because θ n is not the conjugate variable of ϕ n , but Θ n is. To this end, we express the EOMs in Eq. 102 in terms of {Ω k }, which in turn depends on {θ n , ϕ n } (see Eq. B2-Eq. B4). To obtain ∂H s /∂ϕ n , we use the expression of H s in Eq. 86 and explicitly take the derivative with respect to ϕ n . Note that only Ω α jk and Ω β jk contain ϕ n , whereas Ω γ k only contains {θ j }. Using the detailed expressions of Ω α jk (Eq. B2) and Ω β jk (Eq. B3), we have