UM-P-91/45 The classical Schr¨ odinger equation ∗ K. R. W. Jones School of Physics, University of Melbourne, Parkville 3052, Melbourne, Australia. (Dated: Original Preprint: 10 December 1991) Abstract Using a simple geometrical construction based upon the linear action of the Heisenberg–Weyl group we deduce a new nonlinear Schr¨odinger equation that provides an exact dynamic and ener- getic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes is effected in a manner that avoids dispersion and thereby leads to a system of wavefunction dy- namics such that the expectation values track classical trajectories precisely for arbitrarily long times. Interestingly, the value of  proves immaterial for the purpose of constructing this alterna- tive version of classical mechanics. The new feature which  does mediate concerns a surprising embedding of Berry’s phase within ordinary classical mechanics. Some interesting problems are exposed concerning inclusion of the projection postulate within this model nonlinear system and we discover a remarkable route for the recovery of the ordinary linear theory. PACS numbers: 0230, 0365, 0545 * Author’s note: This is the original text of a pre-print that was lost for some twenty years. Over the years, people have asked me for a copy but I had none. This work is not refereed and was never published (since it was lost). Given what I know now, I would write it differently, but place it here under Creative Commons 3.0 - Attribution License under the condition that this footnote is displayed. It is old, but may be of interest for the ideas it explores. Opinions expressed may have been superseded by later works. 1 arXiv:submit/0622900 [quant-ph] 30 Dec 2012