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9
An Introduction to Lie Groups
To prepare for the next chapters, we present some basic facts about Lie
groups. Alternative expositions and additional details can be obtained from
Abraham and Marsden [1978], Olver [1986], and Sattinger and Weaver
[1986]. In particular, in this book we shall require only elementary facts
about the general theory and a knowledge of a few of the more basic groups,
such as the rotation and Euclidean groups.
Here are how some of the basic groups occur in mechanics:
Linear and Angular Momentum. These arise as conserved quantities
associated with the groups of translations and rotations in space.
Rigid Body. Consider a free rigid body rotating about its center of mass,
taken to be the origin. “Free” means that there are no external forces, and
“rigid” means that the distance between any two points of the body is
unchanged during the motion. Consider a point X of the body at time
t = 0, and denote its position at time t by f(X, t). Rigidity of the body
and the assumption of a smooth motion imply that f(X, t)=A(t)X, where
A(t) is a proper rotation, that is, A(t) ∈ SO(3), the proper rotation group
of R3, the 3 × 3 orthogonal matrices with determinant 1. The set SO(3)
will be shown to be a three-dimensional Lie group, and since it describes
any possible position of the body, it serves as the configuration space. The
group SO(3) also plays a dual role of a symmetry group, since the same
physical motion is described if we rotate our coordinate axes. Used as a
symmetry group, SO(3) leads to conservation of angular momentum.
266 9. An Introduction to Lie Groups
Heavy Top. Consider a rigid body moving with a fixed point but un-
der the influence of gravity. This problem still has a configuration space
SO(3), but the symmetry group is only the circle group S1, consisting of
rotations about the direction of gravity. One says that gravity has broken
the symmetry from SO(3) to S1. This