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- W1661494265 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." @default.
- W1661494265 created "2016-06-24" @default.
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- W1661494265 date "2005-11-13" @default.
- W1661494265 modified "2023-10-17" @default.
- W1661494265 title "Isospectral flow and Liouville-Arnold integration in loop algebrast" @default.
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- W1661494265 doi "https://doi.org/10.1007/bfb0021440" @default.
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