Abstract
It has often been said that there is no clear extension of the Courant-Snyder theory to coupled dynamics. In particular, one never sees an extension of the symplectic de Moivre type formula beyond one degree of freedom. In the process of analysing the existence and meaning of such a formula, I discovered quite accidentally that Sands' formalism [1], if expressed in terms of Ripken-like lattice functions germane to the full three dimensional oscillator, gives the exact same result as the more accurate Chao theory [2]. This semi-serious paper displays this elegant connection. It is based in the lattice functions of Ripken [3] as extended by Forest [4]. We review Forest's derivation from the point of view de Moivre's formula extended to three degrees of freedom.
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