By Dave DeFusco
When most people think about math, they picture equations on a chalkboard or numbers on a page. But to mathematicians, math is also a way to describe how physical systems evolve over time, whether that鈥檚 the orbits of planets, motion of low-orbit satellites or energy harvesting devices.
In a recent study, Marian Gidea, professor of mathematical sciences in the Katz School at 麻豆区, along with Rafael de la Llave, a Katz School affiliate research faculty member and emeritus professor at Georgia Tech and Tere M-Seara of the Polytechnic University of Catalonia, explored some of the deepest structures that govern how complex systems evolve. Their work, presented in August at the international conference, Conformal Dynamics and Geometry, in Bordeaux, France, investigates how geometry, topology and dynamics all connect in a special class of systems called conformally symplectic systems.
鈥淭hese systems may sound very theoretical,鈥 said Gidea, 鈥渂ut they show up in many real situations. They describe things like motion with friction, where some energy is always lost, or in finance, where future earnings are adjusted to today鈥檚 dollars. Even in these cases, the system follows strict rules that connect small-scale behavior with the overall structure.鈥
At the heart of the study is an idea called a symplectic structure, a geometric fabric that underlies many physical systems. In perfectly conservative systems, like idealized planetary motion, this symplectic fabric doesn鈥檛 change as things evolve. But in conformally symplectic systems, it can stretch or shrink by a constant factor each time the system moves forward.
The researchers discovered that the stretching factor follows strict rules. It depends on the overall shape and structure of the space the system moves in, which mathematicians call the topology of the phase space. In certain situations, this factor is limited to very specific types of numbers, known as algebraic numbers.
鈥淲hat this means is that geometry and topology are not just background scenery,鈥 said Gidea. 鈥淭hey actively constrain the possible behaviors of the system.鈥
To understand the long-term behavior of a system, mathematicians often look for landmarks in its landscape. One of the most important landmarks is something called a normally hyperbolic invariant manifold, or NHIM. These manifolds act like highways for motion: systems can be attracted to them, repelled from them or travel along them.
The researchers found exact conditions for when these special objects, called NHIMs, take on strong geometric order. It depends on how different parts of the system grow or shrink over time. If the rates balance in certain ways, the structure follows strict geometric rules. If not, it still has some order, just not as complete.
鈥淭his is important because NHIMs organize the dynamics,鈥 said Gidea. 鈥淭hey determine the pathways that orbits can take over long time spans, and our results show how their geometry tightly controls their behavior.鈥
Another tool the researchers studied is called the scattering map. It works like a rule that connects where something came from long ago to where it will end up far in the future, without worrying about every detail in between. This black box approach鈥攆ocusing only on the input and output鈥攈as also been very useful in fields like nuclear physics.
Remarkably, the team showed that these scattering maps retain symplectic properties even when the overall system is dissipative, meaning it loses energy. When the underlying symplectic form is exact, the scattering maps are exact as well.
鈥淭his was a surprising finding,鈥 said Gidea. 鈥淓ven when the system is leaking energy, the scattering maps preserves the energy. That means we can use them to predict how trajectories will behave over very long times.鈥
Though the study is highly theoretical, the insights could influence how scientists and engineers model real-world systems where dissipation and geometry play a role. Gidea is working with Physics Professor Fredy Zypman, students and other collaborators on using the results of this study to design more efficient energy harvesting devices. Some working prototypes already exist. The idea is to use the vibrations in bridges or other structures to power sensors that report on the safety of the structure.
鈥淥ur work shows that you cannot separate the geometry from the dynamics,鈥 said Gidea. 鈥淭hey are deeply intertwined, and by uncovering these connections, we can better understand the universal rules that govern complex systems and make effective predictions or design interesting devices.鈥
Much work still lies ahead. When researchers discover surprising links between fields like geometry and dynamics, it opens a door to many new questions. The next steps are to explore those connections further, develop new mathematical tools and apply them to real-world problems.
鈥淢athematics isn鈥檛 just about studying general systems in a self-selected class,鈥 said de la Llave. 鈥淥ne needs to deal with the systems that are handed to us by nature. The ultimate proof of deep understanding is designing and building a useful new device that did not exist before.鈥
