n!
(This series always converges.) Some easy calculations show that
exp((s+t)A) = exp(sA) exp(tA)
for all s, t ∈R. Also, the entries of the matrix exp(tA) are smooth functions of t∈R.
1. Suppose that Ais skew-adjoint, meaning A∗=−A. Show that exp(tA)∈O(n) for all t∈R.
The group O(n) includes both rotations and reflections. In particular, O(n) consists of two connected
components — the component where det(R) = 1 and the component where det(R) = −1. We define
the rotation group or special orthogonal group SO(n) to be the subgroup consisting of all
R∈O(n) with det(R) = 1. This subgroup only includes rotations. A continuous curve can never
go from one component to another. So, if Ais skew-adjoint, exp(tA) must actually lie in SO(n) for
all t.
We define so(n) to be the set of all skew-adjoint real n×nmatrices. This set so(n) is actually a Lie
algebra, since it is a vector space closed under the bracket operation [x, y] = xy −yx. It is called
the Lie algebra of the rotation group.
Now, let R2nbe the phase space for a particle in Rn. A point (q, p)∈R2ndescribes the particle’s
position q∈Rnand momentum p∈Rn. The algebra of smooth real-valued functions C∞(R2n)
becomes a Poisson algebra with
{F, G}=
n
X
i=1
∂F
∂pi
∂G
∂qi
−∂G
∂pi
∂F
∂qi
.
2. Given A∈so(n), let
φ:R×R2n→R2n
be given by
φ(t, q, p) = (exp(tA)q, exp(tA)p).
Using the facts I’ve told you, show that φis a flow.
1
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