versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 2008
This is a survey of some of the ways in which Quaternions, Octonions and the exceptional group appear in today's Mechanics, addressed to a general audience.
The ultimate reason of this appearance is that quaternionic multiplication turns the 3-sphere of unit quaternions into a group, acting by rotations of the 3-space of purely imaginary quaternions, by
This has been known for quite some time and is perhaps the simplest realization of Hamilton's expectations about the potential of quaternions for physics. One reason for the renewed interest is the fact that the resulting substitution of matrices by quaternions speeds up considerably the numerical calculation of the composition of rotations, their square roots, and other standard operations that must be performed when controlling anything from aircrafts to robots: four cartesian coordinates beat three Euler angles in such tasks.
A more interesting application of the quaternionic formalism is to the motion of two spheres rolling on each other without slipping, i.e., with infinite friction, which we will discuss here. The possible trajectories describe a vector 2-distribution on the 5-fold , which depends on the ratio of the radii and is completely non-integrable unless this ratio is 1. As pointed out by R. Bryant, they are the same as those studied in Cartan's famous 5-variables paper, and contain the following surprise: for all ratios different from 1:3 (and 1:1), the symmetry group is , of dimension 6; when the ratio is 1:3 however, the group is a 14-dimensional exceptional simple Lie group of type .
The quaternions and (split) octonions help to make this evident, through the inclusion
This phenomenon has been variously described as "the 1:3 rolling mystery", "a mere curiosity", "uncanny" and "the first appearance of an exceptional group in real life". Be as it may, it is the subject of current research and speculation. For the history and recent mathematical developments of rolling systems, see [Agrachev][Bor-Montgomery][Bryant-Hsu][Zelenko].
The technological applications deserve a paragraph, given that this Volume is dedicated to the memory of somebody especially preoccupied with the misuse of beautiful scientific discoveries. Quaternions are used to control the flight of aircrafts due to the advantages already cited, and "aircrafts" include guided missiles. A look at the most recent literature reveals that research in the area is being driven largely with the latter in mind. Octonions and , on the other hand, although present in Physics via Joyce manifolds, seem to have had no technological applications so far - neither good nor bad. Still, the main application of Rolling Systems is to Robotics, a field with plenty to offer, of both kinds. The late Misha was rather pesimistic about the chances of the good eventually outweighting the bad. "Given the current state of the world", he said about a year before his death, "the advance of technology appears to be more dangerous than ever".
I would like to thank Andrei Agrachev for introducing me to the subject; John Baez, Gil Bor, Robert Bryant, Robert Montgomery and Igor Zelenko for enlightening exchanges; and the ICTP, for the fruitful and pleasant stay during which I became aquainted with Rolling Systems.
Recall the quaternions,
can also be defined as pairs of complex numbers - much as consists of pairs of real numbers. One sets
Inded, multiplying by a unit quaternion on the left or on the right, is a linear isometry of , as well as conjugating by it
Quaternions themselves come in when fast computation of composition of rotations, or square roots thereof, are needed, as in the control of an aircraft. For this, one needs coordinates for the rotations - three of them, since SO(3) is the group de matrices
To coordinatize one uses the Euler angles, or variations thereof, of a rotation, obtained by writing it as a product where
The configuration space of a pair of adjacent spheres is . Indeed, we can assume one of the spheres to be the unit sphere . Then, the position of the other sphere is given by the point of contact , together with an oriented orthonormal frame attached to . This may be better visualized by substituting momentarely by an aircraft moving over the Earth at a constant height, a system whose configuration space is the same (airplane pilots call the frame the "attitude" of the plane). Identifying with the rotation such that , where is the standard frame in , the configuration is then given by the pair
From now on, we will abandon the use of boldface letters for quaternions.
Replace the configuration space by its 2-fold cover , viewed quaternionically as
Theorem. A rolling trajectory satisfies (NS) and (NT) if and only if , where is tangent to the distribution
Proof: is tangent to if and only if for some smooth , and Eliminating ,
The distribution is integrable if and only if , that is, the spheres have the same radius. Otherwise, it is completely non integrable, of type (2,3,5), meaning that vector fields lying in it satisfy , . These are the subject of E. Cartan's famous "Five Variables paper" and were recognized as rolling systems by R. Bryant. Cartan and Engel provided the first realization of the exceptional group as the group of automorphisms of this differential system for , the connection with "Cayley octaves" being made only later.
Given a vector distribution on a manifold , a global symmetry of it is a diffeomorphism of that carries to itself. They form a group, . But most often one needs local difeomorphisms too, hence the object of interest is really the Lie algebra , but we shall not emphasize the distinction until it becomes significant.
If is integrable, is infinite-dimensional, as can easily be seen by foliating the manifold. At the other end, if is completely non-integrable ("bracket generating"), is generically trivial.
The rolling systems just described all have a symmetry, as can be deduced from the physical set up. More formally, a pair of rotations acts on by
In the covering space , however, the action of extends to an action of a group of type , yielding local diffeomorphisms of the configuration space, as we see next. More precisely,
The realization can be continued recursively to define the sequence of Cayley-Dickson algebras:
There is a split version of these algebras, where the product is obtained by changing the first minus in the formula by a +:
The main contribution of the Cayley-Dickson algebras to mathematics so far has been the fact that the automorphisms of the octonions provide the simplest realization of Lie groups of type . More precisely, the complex Lie group of this type is the group of automorphisms of the complex octonions (i.e., with complex coefficients), its compact real form arises similarly from the ordinary real octonions and a non-compact real form arises from the split one. In physics, the Joyce manifolds of CFT carry, by definition, riemannian metrics with the compact as holonomy, while in rolling it is that matters.
The formula for the product in yields so that for all the distributions can be written as
Lemma: For every octonion ,
a subspace we will denote by .
To prove the Lemma, note that every subalgebra of a generated by two elements is associative (i.e., is "alternative"). Therefore , proving one inclusion. The other uses the quadratic form associated to the split octonions, which also clarifies de action of . It is which on can be replaced by its negative
Now, consider the group a non-compact simple Lie group of type and dimension 14. It fixes . On , which is the orthogonal complement of 1 under , this form is just , which is also preserved by . Hence the quadratic form on all of is -invariant, hence so is . This determines an inclusion
On the configuration space of the rolling system, the elements of act only locally, via the local liftings of the covering map . The local action, of course, still preserves the distribution .
[Agrachev] Agrachev, A. A. Rolling balls and octonions. Proc. Steklov Inst. Math. 258 (2007), no. 1, 13-22 [ Links ]
[Bryant-Hsu] Bryant, Robert L.; Hsu, Lucas. Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 (1993), no. 2, 435-461. [ Links ]
[Zelenko] Zelenko, Igor. On variational approach to differential invariants of rank two distributions. Differential Geom. Appl. 24 (2006), no. 3, 235-259. [ Links ]
[Jacobson] Jacobson, Nathan. Basic algebra. I. W. H. Freeman, San Francisco, Calif., 1974. [ Links ]
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
Recibido: 3 de julio de 2008
Aceptado: 26 de noviembre de 2008