The paper classifies four potentials (inverse-square, oscillator, oscillator plus inverse-square, and Morse) whose Newtonian equations can be embedded in higher-dimensional geometries where the motion is that of a free particle.
Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CR S Press LLC, Florida (2000)
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Linearization of Newton's second law
The paper classifies four potentials (inverse-square, oscillator, oscillator plus inverse-square, and Morse) whose Newtonian equations can be embedded in higher-dimensional geometries where the motion is that of a free particle.