NYC Gazette

Dr. Marian Gidea, director of the Katz School’s graduate program in mathematical sciences, recently presented his research at the 9th Workshop on Hamiltonian Systems and Related Topics held at Kyoto University. His study focuses on instability within a celestial mechanics model known as the Elliptic Hill Four-Body Problem (EH4BP). This work builds upon a theory first introduced by Russian mathematician Vladimir Arnold in 1964.

Hamiltonian systems are used to describe object movements in physics and astronomy, often exhibiting chaotic behavior where small initial changes can lead to significant effects. One such phenomenon is Arnold Diffusion, where energy spreads across a system over time. Dr. Gidea's research delves into autonomous Hamiltonian systems that theoretically preserve energy but can experience shifts due to perturbations.

Collaborating with Jaime Burgos-Garcia from Universidad Autonoma de Coahuila and Claudio Sierpe from University of Bio-Bio, Dr. Gidea examined how four celestial bodies interact when one is significantly smaller than the others. “The model builds on Arnold’s original idea, showing how energy shifts within the system even when external forces remain constant,” said Dr. Gidea.

His research demonstrated how minor variations in larger bodies' movements could cause significant orbital changes for smaller objects using advanced mathematical methods like scattering maps and shadowing lemmas. A real-world example inspired by the Sun-Jupiter-(624) Hektor system was used to illustrate these concepts.

One key aspect of Dr. Gidea's work is understanding perturbations—small influences that push systems from stability to instability—and their role in energy distribution changes within EH4BP and PER4BP models. Transition chains of invariant tori are crucial for Arnold Diffusion, enabling unpredictable energy movement across different regions.

Understanding these instabilities has several implications:

- Space Missions: Trajectory design must consider unpredictable shifts.

- Asteroid Movement: Helps predict long-term asteroid behavior.

- Fundamental Physics: Offers insights into chaotic systems.

- Planetary Defense: Aids in predicting potential asteroid impacts.

- Exoplanet Studies: Improves analysis of exoplanetary system stability.

Dr. Gidea emphasized the importance of computer-assisted proofs for validating numerical experiments, stating, “By using the computer for proving rigorously mathematical facts that are otherwise beyond reach,” we gain deeper insights into celestial dynamics and space exploration.