Geometric Dynamical Systems Evolve from Moscow School Extensions to the Azarbaijan School Foundations

The study of integrable dynamical systems, those exhibiting predictable and non-chaotic behaviour, has undergone a significant evolution, and Ghorbanali Haghighatdoost from Azarbaijan Shahid Madani University, along with colleagues, now charts this progression and presents a novel synthesis of established and modern approaches. Building upon the foundational work of the Moscow School, this research demonstrates how geometric concepts, initially developed for classifying integrable systems through structures like Liouville foliations, have expanded to incorporate advanced mathematical tools such as Lie groupoids and fractional calculus. The team’s work establishes a coherent Iranian school of geometric dynamical systems, bridging classical mechanics with contemporary fields like control theory and advanced mathematical structures, and ultimately revealing the potential for a universal language describing complex systems across diverse scientific disciplines. This achievement highlights a transformation of integrable geometry from a purely mechanical theory into a versatile framework with broad applications.

Integrable Systems, Topology and Bifurcation Analysis

This document details the research interests and activities spanning several decades, centering on integrable Hamiltonian systems and understanding the geometric properties of surfaces defined by constant energy. Researchers systematically classify these systems, often within the framework of Lie algebras, notably so(4), and analyze how their behavior changes as parameters are varied, extending the study of Hamiltonian systems to the more general setting of Lie groupoids, exploring Poisson-Nijenhuis structures and related concepts. Alongside this core research, investigations explore connections to noncommutative geometry, multiplier Hopf algebras, and related algebraic structures, examining properties like the Kazhdan property and cyclic cohomology, underpinned by a broad interest in the topological and geometric aspects of dynamical systems, manifolds, and algebraic structures. A recurring theme is the prominence of the so(4) Lie algebra, indicating a strong focus on systems with related symmetry, and a consistent focus on identifying systems with a large number of conserved quantities. Extending results from Lie algebras to Lie groupoids demonstrates a commitment to bridging the gap between algebraic and geometric approaches to dynamical systems, alongside active participation in conferences, seminars, and workshops, demonstrating a strong commitment to disseminating research and collaboration. In conclusion, this is a highly active and prolific researcher with a long-standing interest in the geometric and algebraic aspects of integrable Hamiltonian systems, characterized by a deep understanding of Lie algebras, Lie groupoids, and related mathematical structures, and a commitment to applying these tools to the study of dynamical systems, demonstrated by an extensive list of publications and presentations.

Geometric Classification of Integrable Hamiltonian Systems

This research builds upon the foundational geometric theory of integrable Hamiltonian systems pioneered by A. T. Fomenko and his Moscow School, extending it into a contemporary framework known as the Azarbaijan School of Geometric Dynamical Systems. The initial stage involved a systematic classification of integrable systems, employing Liouville foliations, atoms, and molecular invariants to move beyond analytical solutions toward a qualitative, geometric understanding of dynamics, replacing complex differential equations with the clarity of geometric structures. Through this method, classical cases of rigid body motion were systematically classified, and their corresponding Liouville foliations were computed to reveal how topology changes with physical parameters. Initiated by A. T. Fomenko’s geometric and topological classification, this approach redefined dynamical systems through concepts such as atoms, molecules, and foliations, offering a new language for understanding stability, bifurcation, and symmetry. This school integrates differential geometry, mathematical physics, fractional calculus, quantum algebra, and non-commutative geometry, expanding the applicability of geometric dynamical systems beyond idealized mechanical systems, providing a framework relevant to control theory, biological modeling, and quantum mechanics, and demonstrating the enduring value of geometric thinking.