Geometric Dynamical Systems

TL;DR

The paper surveys the evolution of integrable Hamiltonian systems from the Moscow School of Fomenko to the Azarbaijan School, emphasizing a shift from analytic solutions to a geometric/topological framework built on Liouville foliations, atoms, and molecules and extended to modern structures like Lie groupoids and algebroids. It details the author's extension to $\mathfrak{so}(4)$, the resulting new topological molecules, and the groundwork for Lie–Poisson/groupoid dynamics, then documents the emergence of a comprehensive Iranian program that integrates topology, geometry, fractional calculus, and quantum algebra. Key contributions include the $\mathfrak{so}(4)$ classification, Euler–Poincaré and Hamilton–Jacobi formulations on groupoids/algebroids, fractional geometric mechanics, and the development of national collaborations and training within the Azarbaijan School. Collectively, these advances position geometric dynamical systems as a universal language connecting topology, control theory, and quantum structures, with wide-ranging implications for mathematical physics, biology, and applied dynamics.

Abstract

This article provides a conceptual and historical review of the evolution of integrable Hamiltonian systems from the Moscow School of A. T. Fomenko to the emerging Azarbaijan School of Geometric Dynamical Systems founded by the author. Beginning with the topological classification of integrable systems through Liouville foliations, atoms, and molecular invariants, the paper traces how these geometric ideas evolved into modern frameworks based on Lie groupoids, Lie algebroids, and fractional calculus. The author s doctoral dissertation at Moscow State University extended 2004 the Fomenko theory to new integrable systems on so(4), constructing novel topological molecules and describing the hierarchy of singularities and bifurcations. Upon his return to Iran, he established a comprehensive research program at Azarbaijan Shahid Madani University, integrating topology, geometry, and analysis to form a coherent Iranian branch of the global theory of integrable systems. This program unifies the classical and the modern: from the Euler Poincare and Hamilton Jacobi formalisms on Lie groupoids and algebroids, to fractional and quantum mechanical models involving Hopf and C algebras. The paper emphasizes conceptual synthesis over computation, showing how integrable geometry has transformed from a purely mechanical theory into a universal language connecting topology, control theory, and quantum structures.