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Dynamics determines geometry. (English) Zbl 1260.81136

Summary: The inverse problem of calculus of variations and \(s\)-equivalence are re-examined by using results obtained from non-commutative geometry ideas. The role played by the structure of the modified Poisson brackets is discussed in a general context and it is argued that classical \(s\)-equivalent systems may be non-equivalent at the quantum mechanical level. This last fact is explicitly discussed comparing different approaches to deal with the Nair-Polychronakos oscillator.

MSC:

81R60 Noncommutative geometry in quantum theory
53D55 Deformation quantization, star products
49N45 Inverse problems in optimal control
35Q55 NLS equations (nonlinear Schrödinger equations)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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