Abrarov Dmitry. Zeta-functional exact general solution of the Euler-poisson equations as a "modular top" and its duality to the one-degree δ-functional dynamic system "vertical pendulum"
Natural Sciences / Mathematics / Dynamical systems
Submitted on: Nov 19, 2024, 18:02:45
Description: The non-classical structure of the exact general solution of the Euler-Poisson equations (EPE), detailed in [1],[2], is demonstrated as an equivariant analytic continuation (globalization) of the phase flow of the Kovalevskaya top at t=∞. The result of this globalization, realized by the hidden symmetry of the involution of the EPE time reversibility, is interpreted as an equivariant Riemann zeta-function with the functional structure of the canonical global I-function in the form of the equivariant correction of the Dirac I-function. The general solution of the EPE has the mechanical meaning of the general EPE-top with a phase flow in the form of a general equivariant perturbation of the flow of the so-called the trivial top. The general solution of the EPE represents the canonical rotation of the 3d sphere along its loxodrome field and integrates to a Ephase-rigid 3d ball BE E" a general EPE top with a paradoxical structure of the phase flow equivalent to the phase flow of a classical pendulum in vertical equilibrium, i.e., it has a formally 1-degree and natural quantum structure in the spirit of the Eloop model for the Navier-Stokes equationsE from [3]. The ball B is the Ecanonical section spaceE of the EPE phase flow, realizing the orbit of its Kowalewskaya-Galois normalization with a 1-dimensional continuous-discrete spectrum is a EModular topE. A EModular topE is a new EuniversalE integrable case of the EPE. In the physical context, a Modular top represents not only the space of self-oscillations of the formally 1-degree Evertical pendulumE, but also its equivalent realizations in the form of a Etetrahedral oscillatorE and a Epurely imaginary oscillatorE. The interpretation of such a E3-1E equivalence as an Eenergy-timeE duality is discussed, and models of gravity and real time are presented in the context of the dynamics of the EPE. A Modular top is a canonical orbit of the symmetry of the time reversib...