Eva Miranda. Penélope Cruz sings in Volver, Pedro Almodóvar's film, Gardel's famous tango: "Back, with withered forehead, the snows of time plated my temple. Feel, that life is a breath, that twenty years is nothing … ", reminding us that returning to the starting point is always possible. This act of returning, so poetic and cinematic, can also be studied mathematically from different points of view.
Periodic phenomena, whose fundamental pattern "back to top", can be associated with an algebraic structure called group, formed by a set and an operation defined on it, that meets certain properties. A simple example would be the clock. Every 12 hours the needle returns to the same place drawing a circumference. The circumference, next to the operation of the translation of the needles would be a group.
On the other hand, periodicity is also a behavior reproduced by some solutions of differential equations; when you draw these solutions you get called circles periodic orbits. For example, in the solar system the trajectories of the planets are solutions of differential equations, which arise when applying the laws of conservation of the energy of the system, described by a scalar function called Hamiltonian.
But is it possible to know the existence and location of periodic orbits in systems of differential equations? This is fundamental, for example, to know if a satellite will return to the starting point or fall in the middle of nowhere. However, due to system disturbances, it is sometimes not easy to calculate the orbits, and to do so it is necessary to use all kinds of techniques: own methods of differential equations but also tools of geometry and topology. In particular, we try to relate the properties of the orbits with the form (or topology) of the so-called phase space (which is the set of all the positions and moments of the system).
One of the mathematicians who made great strides in this problem was the Frenchman Henri Poincaré, also known as the conjecture that bears his name and that was demonstrated by the only person who has rejected the Fields Medal in all its history, Grigori Perelman. Poincaré studied the periodic orbits in the problem of the three bodies, which analyzes the movement of three objects attracted to each other (the Sun, the Earth and the Moon, for example). In this case, the trajectories, described by differential equations, present the additional complexity of being a non-integrable system (that is, we do not have sufficient first integrals to locate their solutions).
Poincaré also worked on the restricted version of the three-body problem (where one of the bodies is assumed to have practically zero mass and the bodies move in a plane), and showed that there are a myriad of periodic orbits that cluster near each other. what is known as the infinity line. The methods used by Poincaré are proposed for concrete systems of celestial mechanics, but they are very effective in many other situations and, in some cases, no other tool that matches or improves their results has yet been found.
In the 70s, the mathematicians Paul Rabinowitz and Eduard Zehnder identified the periodic orbits with the critical points (where the derivative is canceled) of a function called functional action, using variational calculation. A student of Zehnder's thesis, Andreas Floer, went further and related all the critical points of the action functional with an algebraic object (Floer homology). In particular, Floer managed to prove that if one of the Floer homologies associated with the system is not zero then there are periodic orbits. This theory combines tools of topology, geometry, analysis and algebra to solve a problem of dynamic systems.
However, this is just one of the weapons in our arsenal. Sometimes Floer's techniques do not improve the results obtained by Poincaré, but they offer a better understanding of the problem by opening new doors. These more theoretical advances, applied together with own techniques of numerical calculation and dynamic systems allow us to address, among others, problems of celestial mechanics and spatial dynamics not yet solved, like the design of the trajectory of a satellite.
-Alice: "How much time is forever?"
-The white rabbit: "Sometimes just a second."
Lewis Carroll, Alice in Wonderland
Eva Miranda is a professor in Geometry and Topology ICREA Academia at the Polytechnic University of Catalonia, chercheur associé at the Observatory of Paris and a doctor linked to the ICMAT
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".
Editing and coordination: Agate Rudder (ICMAT)