Isospectral flow and Liouville-Arnold integration in loop algebrast

Abstract

Some standard examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schrödinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, embedding certain finite dimensional symplectic manifolds, which arise through Hamiltonian reduction of symplectic vector spaces, into rational coadjoint orbits of loop algebras $\widetilde{\mathfrak{g}}^+$ ⊂ $\widetilde{\mathfrak{g}\mathfrak{l}}(r)^+$.Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps; the isospectral property follows from the Adler-Kostant-Symes theorem. The structure of the generic spectral curves arising through the moment map construction is examined. Spectral Darboux coordinates are introduced on rational coadjoint orbits in $\widetilde{\mathfrak{g}\mathfrak{l}}(r)^{ + *}$, and these are shown to generalize the hyperellipsoidal coordinates encountered in the previous examples. Their relation to the usual algebro-geometric data, consisting of linear flows of line bundles over the spectral curves, is given. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.

Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Fonds FCAR du Québec