Page 33 - Textos de Matemática Vol. 36

2.3. A representation formula 23
One of the questions which led Count Jacopo Francesco Riccati to be- come interested in quadratic differential equations was the following (see the introduction to [BLW91]):
Suppose that a trajectory in the plane satisfies a linear differential equa- tion. What can be said about the time-evolution of the slope of the line through the origin determined by the trajectory?
Consider the linear (possibly with time varying coefficients) differential equation in the (q, p)-plane described by
q˙ m11 m12q
p˙ = m21 m22 p . (2.20)
Let w = p denote the slope of the line through the origin which is determined q
by the point (q, p). It is then straightforward to see that w is a solution of the differential equation
w˙ = m21 + (m22 − m11)w − m12w2. (2.21)
Thus, the slope w evolves as the solution of a Riccati differential equation. This behavior could be generalized to higher dimensions in Radon’s lemma as presented in Theorem 2.2.1 part ii). It should be pointed out here that in the non- commutative case there are other quadratic matrix differential equations which we call generalized Riccati differential equations (also important in applications, where no linearizing transformation, as in Theorem 2.2.1, is known. The study of the behavior of solutions of such equations is much more involved.
Suppose now that instead of the slope p we are interested in the line con- q
necting the point (q, p) with the origin. Let us denote this line [q, p] = [q, p](t). It is no longer an element of R, as the slope w is, but can be considered as an element of the differentiable manifold RP(1), the 1-dimensional projective space; i.e., the manifold consisting of all the lines through the origin in R2.
Since RP(1) is not a Euclidean space, we would need to make a choice of local coordinates before we could write down a differential equation which describes the evolution of [q, p](t). Furthermore, the resulting differential equa- tion would be valid only on some open subset of RP(1) which is homeomorphic (diffeomorphic) to an open subset of R.
However, there is another way to describe the evolution of [q, p](t). Instead of giving a differential equation, we can display the flow explicitly. Let φ(t,t0) denote the transition matrix of the linear system (2.20) which in the case of a
Mt m11 m12 timeinvariantsystemandt0 =0isjuste ,whereM = m m .Let
21 22
q
Im p be identified with the line [q,p] then [q,p](t) evolves according to the