Abstract
A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fast-slow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2-form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.
The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.
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This research was supported by the Higher School of Economics Academic Foundation Project no. 15-01-0146.
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Karasev, M.V. Contribution to the symplectic structure in the quantization rule due to noncommutativity of adiabatic parameters. Russ. J. Math. Phys. 23, 207–218 (2016). https://doi.org/10.1134/S1061920816020060
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DOI: https://doi.org/10.1134/S1061920816020060