Topological analysis and Boolean functions. I. Methods and application to classical systems

TL;DR

This paper introduces a formalized topological analysis method for integrable Hamiltonian systems using Boolean vector-functions, enabling detailed characterization of integral manifolds and oscillation segments.

Contribution

It presents a novel algorithmic approach to analyze the topology of classical integrable systems through Boolean function processing and matrix reduction techniques.

Findings

Defined admissible regions in integral constants space

Characterized segments of oscillation of separated variables

Determined the number of connected components of integral manifolds

Abstract

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.