5 - Lax equations

Summary

Flows on adjoint orbits.

An equation of the form

is called a Lax equation. In the last chapter we considered an example, namely ẋ = [x, Q], where x : R → g, Q ∈ g, and G is a compact Lie group. In this chapter we shall consider the following more general example:

where x, y : R → g, and G is any Lie group. The key to solving this equation (for certain y, at least) is the geometrical property established in the next proposition. We shall denote the adjoint orbit of V by Ov, i.e., O_v = {Ad(_g)V | _g ∈ G} = Ad(G)V.

Proposition. If x is a solution of (⋆), then we have x(t) ∈ O_vfor all t.

Proof. We proved in Chapter 4 that T_xO_v = {[X, x] | X ∈ g}, if G is compact. The same proof is valid if G is non-compact. Therefore, ẋ(= [x, y]) ∈ T_xO_x. It can be deduced from this that x(t) ∈ O_v for all t.▪

Therefore, we may write x(t) = Ad u(t)V(= u(t) Vu(t)^-1), for some u : (–∈, ∈) → G. (This is justified by the fact that the natural map G → G/H = O_v is a locally trivial fibre bundle.) Differentiating the equation x = uVu^-1, we obtain

Comparing this with (⋆), we see that (⋆) is equivalent to the following equation:

Thus, the “change of variable” suggested by the above geometrical property leads to a simplification of the equation.