Global geometry of physical systems and its quantum manifestations
This article appeared in issue 20172 of Periodiek, the magazine of the FMF student association.
Looking at geometric properties of physical systems from a global point of view reveals a rich geometric structure which can be described using ideas from differential geometry and algebraic topology [1]. These geometric properties are relevant to physical systems as fundamental as the hydrogen atom and, even though they arise in the context of classical mechanics, they have significant implications for the corresponding quantum system.
To describe the main ideas that come into play in this research direction I will focus on the example of the spherical pendulum. At first sight, the spherical pendulum is not a very intriguing system. It is integrable, that is, the differential equations describing its motion can be solved in terms of known (but nonelementary) functions.^{1} And you could reasonably ask, if we can find all motions, then don’t we know everything about the system? Well, not quite. Looking at individual motions is like looking through a microscope and may lead to missing the whole picture. There are questions related to global properties which can be answered only if we take a step back and look at the system from a more abstract point of view.
The integrability of the spherical pendulum is not just a happy accident. It has a much deeper reason, related to the existence of two conserved quantities,^{2} the total energy
and the (vertical component of the) angular momentum
Here we are using spherical coordinates where
There are some exceptions to having tori. The most relevant one for this
discussion is the unstable equilibrium, where the spherical pendulum
is motionless at its topmost point with
The next question from a geometric point of view, and a very natural one for a mathematician to ask, is how all the tori fit together. To understand this question, consider the wellknown example of the cylinder and the Möbius band (Figure 4). Even though a cylinder and a Möbius band locally look identical, globally they are very different. Both can be obtained by considering a rectangle, and gluing two opposite sides of the rectangle in different ways.^{3} A consequence of gluing the sides differently is that the cylinder is the Cartesian product of a circle and an interval, while the Möbius band is nonorientable and cannot be written as a Cartesian product.
Back to the spherical pendulum, consider a loop
implying that in this case the bundle is not a Cartesian product and the tori are glued with a “twist”. We say that the spherical pendulum has nontrivial monodromy [2].
These are few of the things that one can say about the global geometry of the spherical pendulum and there are many more interesting connections that one can make with other deep geometric ideas. For example, we have recently shown that the reason for nontrivial monodromy is that a Hopf fibration is hidden in the neighbourhood of the unstable equilibrium [3].
The nontrivial geometry of the spherical pendulum manifests in the
corresponding quantum system. Here one can define the joint spectrum
of the two quantum operators that correspond to the energy and the
angular momentum (Figure 6). You can see in the joint spectrum that if
you take a small cell and you move it along a loop going around
The study of the global geometric properties of physical systems, both at the classical and quantum levels, reveals intriguing geometric structures and explains the structure of the corresponding joint quantum spectra. Here we have only touched on this active field of research in which the group of Dynamical Systems, Geometry, and Mathematical Physics at the JBI is deeply involved.
References
[1] Global aspects of classical integrable systems, 2nd edition. R. H. Cushman and L. M. Bates. Birkhäuser (2015). https://doi.org/10.1007/9783034809184
[2] On global actionangle coordinates. J. J. Duistermaat. Communications on Pure and Applied Mathematics, 33(6):687–706 (1980). https://doi.org/10.1002/cpa.3160330602
[3] Monodromy of Hamiltonian systems with complexity 1 torus actions. K. Efstathiou and N. Martynchuk. Journal of Geometry and Physics, 115:104115 (2017). https://doi.org/10.1016/j.geomphys.2016.05.014
[4] Monodromy in the hydrogen atom in crossed fields. R. H. Cushman and D. A. Sadovskií. Physica D: Nonlinear Phenomena, 142(12):166–196 (2000). https://doi.org/10.1016/S01672789(00)000531
[5] Defect in the joint spectrum of hydrogen due to monodromy. H. Dullin and H. Waalkens. Arxiv preprint (2016). https://arxiv.org/abs/1612.00823

Notice the contrast with chaotic systems where it is not possible to solve the equations of motion. ↩

And there is an even deeper reason: the conservation of the energy and the angular momentum is related to symmetries of the system through Noether’s theorem. ↩

A very similar example is given by the torus and the Klein bottle where the rectangle is replaced by a cylinder and the two circles at the ends of the cylinder are glued in different ways. ↩