Abrarov Dmitry. The fundamental counterexample to Kam-theory: non-existence of toroidal Kam-dynamics and equivariant correction of Kam-chaos
Natural Sciences / Mathematics / Dynamical systems
Submitted on: Feb 25, 2024, 10:51:17
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. The counterexample is based on the symmetry of time reversibility for Hamiltonian systems and induced separatrix dynamics. It has a geometric interpretation of the canonical analytical structure on the three-dimensional sphere. The fundamentality of the counterexample is due to the need for an axiomatic projective extension of the affine structure of KAM-dynamics to the projective structure of the canonical meromorphic extension of the three-dimensional Lobachevsky space. This extension is canonical and provides a correct (equivariant) description of the original affine Hamiltonian KAM-dynamics. This counterexample shows the non-existence of classical KAM-tori and KAM-dynamics in the general situation. By generality here we mean non-degeneracy in the natural sense of homotopy non-triviality of the Liouville foliations for integrable Hamiltonian dynamics. At the same time, the counterexample simply supplements the classical Liouville-Arnold theorem with the case of a continuous smoothness class of phase dynamics, which was omitted in the classical consideration (noted in [2]), and implements the dynamic interpretation of the modular parameterizability of elliptic curves with rational coefficients. This continuous extension gives an equivariant Galois-solvable renormalization of KAM-theory into the conjectural category of analytically integrable Hamiltonian systems. In the case of three degrees of freedom, equivariant renormalization is realized by the special zeta-functional structure and is related to the Kowalewskaya method in the dynamics of classical tops.