Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part I One-dimensional integrable systems
- 1 Lie groups
- 2 Lie Algebras
- 3 Factorizations and homogeneous spaces
- 4 Hamilton's equations and Hamiltonian systems
- 5 Lax equations
- 6 Adler-Kostant-Symes
- 7 Adler-Kostant-Symes (continued)
- 8 Concluding remarks on one-dimensional Lax equations
- Part II Two-dimensional integrable systems
- Part III One-dimensional and two-dimensional integrable systems
- References
- Index
5 - Lax equations
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part I One-dimensional integrable systems
- 1 Lie groups
- 2 Lie Algebras
- 3 Factorizations and homogeneous spaces
- 4 Hamilton's equations and Hamiltonian systems
- 5 Lax equations
- 6 Adler-Kostant-Symes
- 7 Adler-Kostant-Symes (continued)
- 8 Concluding remarks on one-dimensional Lax equations
- Part II Two-dimensional integrable systems
- Part III One-dimensional and two-dimensional integrable systems
- References
- Index
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., Ov = {Ad(g)V | g ∈ G} = Ad(G)V.
Proposition. If x is a solution of (⋆), then we have x(t) ∈ Ovfor all t.
Proof. We proved in Chapter 4 that TxOv = {[X, x] | X ∈ g}, if G is compact. The same proof is valid if G is non-compact. Therefore, ẋ(= [x, y]) ∈ TxOx. It can be deduced from this that x(t) ∈ Ov 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 = Ov 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.
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- Harmonic Maps, Loop Groups, and Integrable Systems , pp. 24 - 30Publisher: Cambridge University PressPrint publication year: 1997