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The Geometric Nature of the Flaschka Transformation

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Abstract

We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable Toda lattice system, is the inverse of a momentum map. We discuss the geometrical setting of the map and apply it to the generalized Toda lattice systems on semisimple Lie algebras, the rigid body system on Toda orbits, and to coadjoint orbits of semidirect products groups. In addition, we develop an infinite-dimensional generalization for the group of area preserving diffeomorphisms of the annulus and apply it to the analysis of the dispersionless Toda lattice PDE and the solvable rigid body PDE.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan Ann Arbor, Ann Arbor, MI, 48109, USA

    Anthony M. Bloch

  2. LMD-IPSL, CNRS-Ecole Normale Supérieure, 75005, Paris, France

    François Gay-Balmaz

  3. School of Mathematics, Department of Applied Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai, 200240, China

    Tudor S. Ratiu

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Tudor S. Ratiu

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  1. Anthony M. Bloch
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  2. François Gay-Balmaz
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  3. Tudor S. Ratiu
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Correspondence to Anthony M. Bloch.

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Communicated by P. Deift

Anthony M. Bloch: Research partially supported by NSF Grants INSPIRE-1363720 and DMS-1207693, DMS-1613819 and the Simons Foundation.

François Gay-Balmaz: Research partially supported by the ANR project GEOMFLUID (ANR-14-CE23-0002-01).

Tudor S. Ratiu: Research partially supported by NCCR SwissMAP grant of the Swiss National Science Foundation.

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Bloch, A.M., Gay-Balmaz, F. & Ratiu, T.S. The Geometric Nature of the Flaschka Transformation. Commun. Math. Phys. 352, 457–517 (2017). https://doi.org/10.1007/s00220-017-2854-5

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