Note that Eq. 117 are the classical Hamilton's EOMs of H (Eq. 118), with the conjugated variables {q n , p n }. Similarly, Eq. 104 is also the Hamilton's EOMs of H s , with the conjugate variables {Θ n , ϕ n }. Thus, the transformation that convert H s (Θ n , ϕ n ) (Eq. 103) to H(q n , p n ) (Eq. 118) is a canonical transform that preserves the form of Hamilton's EOMs. More specifically, this canonical transformation that connects the generalized conjugate variables {Θ n , ϕ n } to the conjugate position and momentum is expressed as