Page 34 - Textos de Matemática Vol. 36

24
flow Im p(t)
Chapter 2.
, which we also call a Riccati flow, where
Global aspects
(2.22)
q(t)
q q0
S(t) = p (t) := φ(t,t0) p q(t)
.
There are several features of the flow Im p(t) worth noting. Firstly, this flow can be viewed as a composition of a projection map
Π : R2 → RP(1), q q
p →[q,p]=Π p q q0
the line [q, p](t) is not vertical.
Thirdly, note that the flow on RP(1) is defined for all time, whereas the
solution w(t) of the Riccati differential equation (2.21) can escape to infinity in finite time. This happens when q(t) first becomes zero or, equivalently, when the line [q, p](t) first becomes vertical.
Some comments about the structure of RP(1) are useful. Each point in RP(1) is a line through the origin in R2, and each such line can be indexed by the angle it makes with the q-axis. This angle varies from 0 to π. However, the angles 0 and π represent the same line, namely the q-axis. Thus, topologically RP(1) can be regarded as the closed interval [0,π] with the two endpoints identified, i.e., the circle S1. R can be embedded in RP(1) by the mapping
ψ : R→RP(1)
w → ψ(w) = [1, w].
The image of this mapping is the subset of RP(1) which consists of all lines which are not vertical. If [q, p] represents such a line (q ̸= 0), then
ψ−1([q,p]) = ψ−1([1, p]) = p. qq
By this embedding, we may identify R with the open and dense subset (actually its complement contains one point, i.e., it has in particular measure zero)
where
q(t)
p . 0
p (t) = φ(t,t0)
[q, p](t) on RP(1) by the formula w(t) = p(t) as long as q(t) ̸= 0; i.e., as long as
with the
Secondly, the trajectory of the slope w(t) is easily recovered from the flow
0
ψ(R) = RP(1) \ [0, 1] ⊂ RP(1).