Abrarov Dmitry. Analytical supersymmetry of the Kowalewskaya top as a key counterexample to the Kam-theory and as a tool for its global zeta-correction based on the real-time model
Natural Sciences / Mathematics / Dynamical systems
Submitted on: Apr 14, 2024, 17:14:32
Description: Based on the zeta-functional structure of the general solution of the Euler-Poisson equations from [1], [2], a geometrically and physically meaningful constructive counterexample to KAM-theory is constructed in the form of Euniversal enveloping KAM-dynamicsE, which is canonically generated by the dynamics of the Kowalewskaya top. The general idea of the counterexample is that toroidal KAM-dynamics is always tautologically embedded in the canonical exponential map of the equivariant extension of three-dimensional Lobachevsky space. This embedding leads to a physically meaningful special zeta-functional reparameterization of KAM dynamics based on a real-time model. This mapping has a paradoxical mechanical interpretation in the form of the phase flow of a classical mathematical pendulum strictly in vertical equilibrium, conjecturally described in complex time by the Painlevé-VI equation. Also, this mapping has a dynamically attractor and algebraic Galois-solvable structure; is analytically supersymmetric and turns out to be equivalent to the phase flow of the Kowalewskaya top; parameterized by the real time-model in the context of the Grebenikov-Demina-Aksenov model of the Earth's gravitational potential. The phase flow of the Kowalewskaya top implements the canonical analytical continuation of the classical affine toroidal dynamics of integrable tops to the points of their fixation and axes of their own rotation; this flow represents the canonical normal form of the Euler-Poisson equations and embodying the mechanical meaning of the Painlevé-Kowalewskaya method. The given counterexample also supplements the classical Liouville-Arnold theorem with the case of a continuous class of smoothness of phase dynamics, which was omitted in the classical consideration. It also implements a dynamic interpretation of the fundamental property of modular parameterizability of elliptic curves with rational coefficients. An inductive procedure based on the dynamics of the ...