Abstract
Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part\(\tilde{\mathfrak{g}}^+\) of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equation. Invent. math.50, 219–248 (1979)
[AHH1] Adams, M.R., Harnad, J., Hurtubise, J.: Isospectral Hamiltonian flows in finite and infinite dimension. II. Integration of flows. Commun. Math. Phys.134, 555–585 (1990)
[AHH2] Adams, M.R., Harnad, J., Hurtubise, J.: Dual moment maps to loop algebras. Lett. Math. Phys.20, 294–308 (1990)
[AHH3] Adams, M.R., Harnad, J., Hurtubise, J.: Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras. In: Hamiltonian systems, transformation groups and spectral transform methods. Harnad, J., Marsden, J. (eds.). Montréal: Publ. C.R.M. 1990
[AHH4] Adams, M.R., Harnad, J., Hurtubise, J.: Liouville generating function for isospectral Hamiltonian flow in loop algebras. In: Integrable and superintegrable systems. Kuperschmidt, B. (ed.) Singapore: World Scientific 1990
[AHH5] Adams, M.R., Harnad, J., Hurtubise, J.: Coadjoint orbits, spectral curves and Darboux coordinates. In: The geometry of Hamiltonian systems. Ratiu, T. (ed.). New York: Publ. MSRI, Springer 1991
[AHH6] Adams, M.R., Harnad, J., Hurtubise, J.: Coadjoint orbits and algebraic geometry (in preparation)
[AHP] Adams, M.R., Harnad, J., Previato, E.: Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalised Moser systems and moment maps into loops algebras. Commun. Math. Phys.117, 451–500 (1988)
[AvM] Adler, M., Van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.38, 267–317 (1980); Linearization of Hamiltonian systems, Jacobi Varieties and Representation Theory. Adv. Math.38, 318–379 (1980)
[C] Clebsch, A.: Leçons sur la géométrie, Vol. 3, Paris: Gauthier-Villars 1883
[D] Dickey, L.A.: Integrable nonlinear equations and Liouville's theorem. I, II. Commun. Math. Phys.82, 345–360 (1981); Commun. Math. Phys.82, 360–375 (1981)
[Du] Dubrovin, B.A.: Theta functions and nonlinear equations. Russ. Math. Surv.36, 11–92 (1981)
[F] Flaschka, H.: Toward an algebro-geometrical interpretation of the Neumann system. Tohoku Math. J.37, 407–426 (1984)
[FRS] Frenkel, I.B., Reiman, A.G., Semenov-Tian-Shansky, M.A.: Graded Lie algebras and completely integrable Hamiltonian systems. Sov. Math. Doklady20, 811–814 (1979)
[GHHW] Gagnon, L., Harnad, J., Hurtubise, J., Winternitz, P.: Abelian integrals and the reduction method for an integrable Hamiltonian system. J. Math. Phys.26, 1605–1612 (1985)
[GD] Gel'fand, I.M., Dik'ii, L.A.: Integrable nonlinear equations and the Liouville theorem. Funct. Anal. Appl.13, 8–20 (1979)
[GH] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978
[H] Harnad, J.: Uses of infinite dimensional moment maps. In: Proceedings of XVIIth International colloquium on Group Theoretical Methods in Physics. Saint-Aubin, Y., Vinet, L. (eds.), Singapore: World Scientific 1989, pp. 68–89
[Kn] Knörrer, H.: Geodesics on quadrics and a mechanical problem of C. Neumann. J. Reine Angew. Math.334, 69–78 (1982)
[K] Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv.32, 185–213 (1977)
[KN] Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and nonlinear equations. Russ. Math. Surv.32, 53–79 (1980)
[Ko] Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math.34, 195–338 (1979)
[vMM] van Moerbeke, P., Mumford, D.: The spectrum of difference operators and algebraic curves. Acta Math.134, 93–154 (1979)
[M] Moser, J.: Geometry of quadrics and spectral theory. The Chern Symposium, Berkeley, June 1979, 147–188, New York: Springer 1980
[N] Neumann, C.: De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur. J. Reine Angew. Math.56, 46–63 (1859)
[Ra] Ratiu, T.: The C. Neumann problem as a completely integrable system on an adjoint orbit. Trans. A.M.S.439, 321–329 (1981)
[RS] Reiman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and lax equations. I, II. Invent. Math.54, 81–100 (1979); Invent. Math.63, 423–432 (1981)
[S] Symes, W.: Systems of Toda type, inverse spectral problems and representation theroy. Invent. Math.59, 13–51 (1980)
[Sch] Schilling, R.: Generalizations of the Neumann system—a curve theoretic approach. Commun. Pure Appl. Math.40, 455–522 (1987)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army Grant DAA L03-87-K-0110
Rights and permissions
About this article
Cite this article
Adams, M.R., Harnad, J. & Hurtubise, J. Darboux coordinates and Liouville-Arnold integration in loop algebras. Commun.Math. Phys. 155, 385–413 (1993). https://doi.org/10.1007/BF02097398
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02097398