In order to obtain equivalent EOMs in terms of the 2N -2 variables, we want to find a set of canonical variables. We recognize that in the two-state special case, the conjugate momentum of ϕ 1 is 1 2 r s cos θ 1 ≡ r s Ω γ2 , where γ 2 is the index of the diagonal generator (see Eq. 5c). Hence, we expect a conjugate momentum of any ϕ n being a combination of {Ω γj } for j ∈ {2, • • • , N }, which to the best of our knowledge, is unknown in the literature. Based on this conjectured conjugate relationship, we postulate the following Hamilton's EOMs