Quaternions and Octonions in Mechanics

Abstract

This is a survey of some of the ways in which Quaternions, Octonions and the exceptional group G2 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 v 7→ qvq−1. In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of

Full text

REVISTA DE LA
UNI ´
ON MATEM´
ATICA ARGENTINA
Volumen 49, N´umero 2, 2008, P´aginas 45–53
QUATERNIONS AND OCTONIONS IN MECHANICS
AROLDO KAPLAN
1. Introduction
This is a survey of some of the ways in which Quaternions, Octonions and the
exceptional group G2appear 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
v7→ qvq−1.
In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of
R3.
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 rea-
son for the renewed interest is the fact that the resulting substitution of matrices
by quaternions speeds up considerably the numerical calculation of the compo-
sition 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 S2×S3, 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 SO(4),
of dimension 6; when the ratio is 1:3 however, the group is a 14-dimensional ex-
ceptional simple Lie group of type G2.
The quaternions Hand (split) octonions Oshelp to make this evident, through
the inclusion
S2×S3֒→ ℑ(H)×H=ℑ(Os).
The distributions themselves can be described in terms of pairs of quaternions, a
description that becomes “algebraic over Os” in the 1:3 case. As a consequence,
Aut(Os), which is preciely that exceptional group, acts by symmetries of the sys-
tem.
This phenomenon has been variously described as “the 1:3 rolling mystery”,
“a mere curiosity”, “uncanny” and “the first appearance of an exceptional group
45

46 AROLDO KAPLAN
in real life”. Be as it may, it is the subject of current research and specula-
tion. 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 G2, 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 sub ject; John
Baez, Gil Bor, Robert Bryant, Robert Montgomery and Igor Zelenko for enlight-
ening exchanges; and the ICTP, for the fruitful and pleasant stay during which I
became aquainted with Rolling Systems.
2. Quaternions and Rotations
Recall the quaternions,
H={a=ao+a1i+a2j+a3k:aj∈R}∼
=R4
as a real vector space, endowed with the bilinear multiplication ab defined by the
relations
i2=j2=k2=−1ij =k=−ji jk =i=−kj ki =j=−ik.
His an associative algebra, like Ror C, where every non-zero element has an
inverse, satisfying a−1(ab) = b= (ba)a−1, i.e., it is a division algebra. But unlike
Ror C, it is clearly not commutative.
Hcan also be defined as pairs of complex numbers – much as Cconsists of pairs
of real numbers. One sets
H=C×C
with product
(A, B)(C, D) = (AC −D¯
B, ¯
AD +CB).
Under the equivalence, i= (0,1), j= (i, 0), k= (0, i),and the conjugation
ao+a1i+a2j+a3k=ao−a1i−a2j−a3k
becomes
(A, B) = ( ¯
A, −B).
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QUATERNIONS AND OCTONIONS IN MECHANICS 47
The formula ab= (ao+a1i+a2j+a3k)(bo−b1i−b2j−b3k) = (aobo+a1b1+
a2b2+a3b3) + (...)i+ (...)j+ (...)kshows that, just as in the case of R2=C, the
euclidean inner product in R4=Hand the corresponding norm are
<a,b>=ℜ(ab),|a|2=aa.
Since |ab|=|a||b|,
a−1=a
|a|2.
The role of quaternions in mechanics comes through identifying euclidean 3-space
with the imaginary quaternions ℑ(H) (= span of i,j,k) and the following fact:
under quaternionic multiplication, the unit 3-sphere
S3={a∈H:|a|= 1}
is a group, and the map
S3× ℑ(H)→ ℑ(H),(a,b)7→ aba−1
is an action of this group by rotations of 3-space.
Inded, multiplying by a unit quaternion aon the left or on the right, is a linear
isometry of H∼
=R4, as well as conjugating by it
ρa(b) = aba−1=ab¯
a.
The transformation ρapreserves ℑ(H), since for an imaginary b,ρa(b) = ab¯
a=
¯
¯
a¯
b¯
a=a¯
b¯
a=−ab¯
a=−ρa(b).In fact, ρais isometry of ℑ(H) = R3, i.e., an element
of the orthogonal group O(3). Indeed, ρ(S3) = SO(3), because S3is compact and
connected, and
ρ:S3→SO(3)
is a Lie group homomorphism. This is a 2-1 map:
ρa=ρ−a.
In fact,
Z2֒→S3→SO(3)
is the universal cover of SO(3). In particular, the fundamental group of the rotation
group is
π1(SO(3)) = Z2.
This “topological anomaly” of 3-space has been noted for a long time, and used
too: if it wasn’t for it, there could be no rotating bodies – wheels, centrifuges, or
turbines – fed by pipes or wires connected to the outside. In practice, by turning
the latter twice for every turn of the body, the resulting “double twist” can be
undone by translations.
Quaternions themselves come in when fast computation of composition of rota-
tions, or square roots thereof, are needed, as in the control of an aircraft. For this,
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48 AROLDO KAPLAN
one needs coordinates for the rotations – three of them, since SO(3) is the group
de matrices
A=

a11 a12 a13
a21 a22 a23
a31 a32 a33

AtA=I, det A= 1
and 9 parameters minus 6 equations leave 3 free parameters.
To coordinatize SO(3) one uses the Euler angles, or variations thereof, of a rotation,
obtained by writing it as a product EαFβEγwhere
Eα=

cos αsin α0
−sin αcos α0
0 0 1

, Fα=

1 0 0
0 cos αsin α
0−sin αcos α

.
But in
(Eα1Fβ1Eγ1)(Eα2Fβ2Eγ2) = Eα3Fβ3Eγ3
(Eα1Fβ1Eγ1)1/2=Eα3Fβ3Eγ3
the funcions α3, β3, γ3are complicated expressions in α1, β1, γ1, α2, β2, γ2. Fur-
thermore, when large rotations are involved, the multivaluedness and singularities
of the Euler angles also lead to what numerical programmers know as “computa-
tional glitch”. Instead, S3is easier to coordinatize, the formula for the quaternionic
product is quadratic, and for |a|= 1, a6=−1,
√a=1
p2(ℜ(a) + 1) a+ 1
The price paid by these simplifications is the need of the non-linear condition
|a|2= 1. There is an extensive recent literature assesing the relative computational
advantages of each representation, easily found in the web.
3. Rolling spheres
The configuration space of a pair of adjacent spheres is S2×SO(3). Indeed, we
can assume one of the spheres Σ1to be the unit sphere S2⊂R3. Then, the position
of the other sphere Σris given by the point of contact q∈S2, together with an
oriented orthonormal frame Fattached to Σr. This may be better visualized by
substituting momentarely Σrby an aircraft moving over the Earth Σ1at a constant
height, a system whose configuration space is the same (airplane pilots call the
frame Fthe “attitude” of the plane). Identifying Fwith the rotation ρ∈SO(3)
such that ρ(Fo) = F, where Fois the standard frame in R3, the configuration is
then given by the pair
(q, ρ)∈S2×SO(3).
Now let Σrroll on Σ1describing the curve (q(t), ρ(t)) ∈S2×SO(3). The non-
slipping condition is encoded into two equations, expressing the vanishing of the
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QUATERNIONS AND OCTONIONS IN MECHANICS 49
linear and of the angular components of the slipping (“no slipping or twisting”),
namely
(NS) (1 + 1
r)q′=ρ′ρ−1(q)
(N T )ω⊥q
where ω×v=ρ′ρ−1(v) is the angular velocity of Σr(t) relative to the fixed frame
Fo. (NS) says that the linear velocity of the point of contact on the fixed Σ1is the
same as the velocity of the point of contact on Σr(t):
q′(t) = ρ(t)(ρ(t)−1(−rq(t)))′
The right-hand side is just the formula for transforming between rotating frames,
given that the point of contact on Σr(t) relative to the fixed frame Fois −rq(t)
plus a translation. Explicitely, relative to the frame Fr(t) this point is (dropping
the t’s) −rρ−1(q) and moves with velocity
(ρ−1(−rq))′=rρ−1ρ′ρ−1q−rρ−1q′.
When rotated back to its actual position in R3, i.e., relative to the frame Fo, it
becomes
ρ(ρ−1(−rq))′=rρ′ρ−1q−rq′.
which is the same as q′=rρ′ρ−1q−rq′, or
1 + r
rq′=ρ′ρ−1(q),
as claimed. (NT) is clearer, stating that Σrcan rotate only about the axis perpen-
dicular to the direction of motion and, because of (NS), tangent to Σ1.
4. Rolling with quaternions
From now on, we will abandon the use of boldface letters for quaternions.
Replace the configuration space S2×SO(3) by its 2-fold cover S2×S3, viewed
quaternionically as
S2×S3֒→ ℑ(H)×H,
and recall the map Q→ρQfrom S3to SO(3), ρQ(v) = QvQ−1. Clearly,
T(qo,Qo)(S2×S3)∼
={(p, P )∈ ℑ(H)×H:< p, qo>= 0 =< P, Qo>}
Theorem.A rolling trajectory (q(t), ρ(t)) ∈S2×SO(3) satisfies (NS) and (NT)
if and only if ρ(t) = ρq(t)Q(t), where (q(t), Q(t)) ∈S2×S3is tangent to the
distribution
D(r)
(qo,Qo)={(qox, 1−r
2rxQo) : x∈q⊥
o⊂ ℑ(H)}.
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50 AROLDO KAPLAN
Proof: (p(t), Q(t)) is tangent to D(r)if and only if for some smooth x=x(t)∈
p(t)⊥∩ ℑ(H), p′=px and Q′=1−r
2rxQ. Eliminating x,
(∗)1−r
2rp′=pQ′Q−1
For a fixed v,
ρ(t)(v) = (p(t)Q(t))v(p(t)Q(t))−1=−pQv ¯
Qp,
and ρ−1(v) = −¯
QpvpQ,ρ′(v) = −(p′Q+pQ′)v¯
Qp −pQv(¯
Q′p+¯
Qp′), and therefore
ρ′(ρ−1(v)) = −p′pv −pQ′¯
Qpv −vpQ ¯
Q′p−vpp′. In particular,
ρ′(ρ−1(p)) = 2p′+pQ′¯
Q+Q¯
Q′p
The (NS)-condition for (x(t), ρ(t)) is then
1 + r
rp′=ρ′ρ−1(p) = 2p′+pQ′¯
Q+Q¯
Q′p.
Since x⊥p, so is Q′Q−1=Q′¯
Qand, because pis purely imaginary, pQ′¯
Q=Q¯
Q′p.
We conclude that the last equation is the same as 1−r
rp′=−2pQ′¯
Q, as claimed.
The rest of the proof proceeds along the same lines.
The distribution D=D(r)is integrable if and only if r= 1, 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 dim{X+ [Y , Z]}= 3,
dim{X+ [Y, Z ] + [U, [V, W ]]}= 5. These are the subject of E. Cartan’s famous
“Five Variables paper” and were recognized as rolling systems by R. Bryant. Car-
tan and Engel provided the first realization of the exceptional group G2as the
group of automorphisms of this differential system for r= 3,1/3, the connection
with “Cayley octaves” being made only later.
5. Symmetries
Given a vector distribution Don a manifold M, a global symmetry of it is a
diffeomorphism of Mthat carries Dto itself. They form a group, Sym(M, D).
But most often one needs local difeomorphisms too, hence the object of interest is
really the Lie algebra sym(M, D ), but we shall not emphasize the distinction until
it becomes significant.
If Dis integrable, sym(M, D) is infinite-dimensional, as can easily be seen by
foliating the manifold. At the other end, if Dis completely non-integrable (“bracket
generating”), sym(M , D) is generically trivial.
The rolling systems just described all have a SO(4) = SO(3)×SO(3) symmetry,
as can be deduced from the physical set up. More formally, a pair of rotations
(g1, g2) acts on S2×SO(3) by
(g1, g2)·(x, ρ) = (g1x, g1ρg−1
2)
This action preserves each of the D(r)’s and are clearly global. Indeed, these are
the only global symmetries that these distributions have for any r6= 1.
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QUATERNIONS AND OCTONIONS IN MECHANICS 51
In the covering space S2×S3, however, the action of SO(4) extends to an action
of a group of type G2, yielding local diffeomorphisms of the configuration space, as
we see next. More precisely,
S2×S3=SO(4)/SO(2) = G′
2/P
where Pis maximal parabolic. However, the lifted distributions D(r)themselves
are not left invariant under the G2-action – except in the case r= 1/3.
6. Octonions
The realization H=C×Ccan be continued recursively to define the sequence
of Cayley-Dickson algebras:
AN+1 =AN×AN
with product and conjugation
(A, B)(C, D) = (AC −D¯
B, ¯
AD +CB)(A, B) = ( ¯
A, −B).
Starting with A0=R,
A1=C,A2=H,A3=O,
the algebra of octonions, which is non-associative. These four give essentially all
division algebras /R; from A4– the sedenions – on, they have zero divisors, i.e.,
nonzero elements a,bsuch that ab = 0.
There is a split version of these algebras, where the product is obtained by
changing the first minus in the formula by a +:
(A, B)(C, D) = (AC +D¯
B, ¯
AD +CB).
Both the standard and the split versions can be expressed as direct sums
AN+1 =AN⊕ℓAN
where ℓ2=±1 according if it is the split one or not. Note that in a split algebra,
(1 + ℓ)(1 −ℓ= 0, hence they have zero divisors from the start.
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 G2. 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 G′
2arises from the split one. In physics, the Joyce
manifolds of CFT carry, by definition, riemannian metrics with the compact G2as
holonomy, while in rolling it is G′
2=Aut(Os) that matters.
Since
ℑ(Os) = ℑ(H)×H∼
=ℑ(H)⊕ℓH
we can write
S2×S3={a=q+ℓQ ∈ ℑ(Os) : |q|= 1 = |Q|}.
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52 AROLDO KAPLAN
These split octonions all have square zero: (p+ℓQ)2= (pp +Q¯
Q) + ℓ(−pQ +pQ).
Indeed, every imaginary split octonion a6= 0 satisfying a2= 0, is a positive multiple
of one in S2×S3.
The formula for the product in Osyields ℓ(xQ) = (ℓQ)xso that for all rthe
distributions can be written as
D(r)
(q+ℓQ)={b=q+1−r
2rℓQx∈ ℑ(Os) : x∈q⊥⊂ ℑ(H)}
In particular, D1/3
a={ax:x∈q⊥⊂ ℑ(H)}. This expression is still not all
octonionic, but its canonical extension to a 3-distribution ˜
Da=D1/3
a+Raon the
cone R+(S2×S3) is:
Lemma:For every octonion a∈S2×S3,
D1/3
a+Ra={b∈ ℑ(Os) : ab = 0},
a subspace we will denote by Za.
To prove the Lemma, note that every subalgebra of a Osgenerated by two
elements is associative (i.e., Osis “alternative”). Therefore a(ax) = a2x= 0,
proving one inclusion. The other uses the quadratic form associated to the split
octonions, which also clarifies de action of Aut(Os). It is ℜ(ab) which on ℑ(Os)) ∼
=
R7can be replaced by its negative
<a,b>=ℜ(ab).
This is a symmetric and non-degenerate, of signature (3,4) – in contrast to the one
for ordinary Octonions, which is positive definite. Moreover, a2= 0 ⇔<a,a>= 0
for imaginary a. It follows that R+(S2×S3) is the null cone of the quadratic form,
and the same as the set of elements of square zero in ℑ(Os). It is now easy to see
that if a2= 0 and ab = 0, then b=λa+axwith xas required.
Now, consider the group G=Aut(Os),a non-compact simple Lie group of type
G2and dimension 14. It fixes 1. On ℑ(Os), which is the orthogonal complement
of 1 under <a,b>, this form is just −ℜ(ab), which is also preserved by G. Hence
the quadratic form on all of Osis G-invariant, hence so is ℑ(Os). This determines
an inclusion
G⊂SO(3,4).
In particular, Gacts linearly on the null cone of the form there. This action
descends to a non-linear, transitive action on S2×S3– much like the action of
SL(n, R) on Rndescends to one on Sn−1. Since g(Za) = Zg(a), the action preserves
the descended Za’s, which are just the fibers of the distribution D. Hence
Aut(Os)⊂Aut(S2×S3, D)
In fact, the two sides are equal.
On the configuration space of the rolling system, the elements of Gact only
locally, via the local liftings of the covering map S2×S3→S2×SO(3). The local
action, of course, still preserves the distribution D1/3.
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QUATERNIONS AND OCTONIONS IN MECHANICS 53
References
[Agrachev] Agrachev, A. A. Rolling balls and octonions. Proc. Steklov Inst. Math. 258
(2007), no. 1, 13–22
[Bor-Montgomery] Bor, Gil; Montgomery, Richard. G2and the “Rolling Distribution”.
arXiv:math/0612469v1 [math.DG], 2006.
[Bryant-Hsu] Bryant, Robert L.; Hsu, Lucas. Rigidity of integral curves of rank 2distribu-
tions. Invent. Math. 114 (1993), no. 2, 435–461.
[Zelenko] Zelenko, Igor. On variational approach to differential invariants of rank two
distributions. Differential Geom. Appl. 24 (2006), no. 3, 235–259.
[Jacobson] Jacobson, Nathan. Basic algebra. I. W. H. Freeman, San Francisco, Calif.,
1974.
Aroldo Kaplan
CIEM-FaMAF,
Universidad Nacional de C´ordoba,
C´ordoba 5000, Argentina
aroldokaplan@gmail.com
Recibido: 3 de julio de 2008
Aceptado: 26 de noviembre de 2008
Rev. Un. Mat. Argentina, Vol 49-2