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Integrable Hamiltonian systems and symmetric products of curves

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1638)

Abstract

This chapter is entirely devoted to the construction and a geometric study of a big family of integrable Hamiltonian systems. The phase space is C2d but equipped with many different Poisson structures: for each non-zero φ ∈ C[x, y] we construct (in Paragraph 2.2) a Poisson bracket {·, ·} φ d which makes (C2d, {·, ·} φ d ) into an affine Poisson variety. Each of these brackets has maximal rank 2d (in particular the algebra of Casimirs is trivial) and they are all compatible. An explicit formula for all these brackets is given; they grow in complexity (i.e., degree) with φ so that only the first members are (modified) Lie-Poisson structures.

Keywords

  • Vector Field
  • Poisson Bracket
  • Poisson Structure
  • Integrable Hamiltonian System
  • Symmetric Product

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1996 Springer-Verlag Berlin Heidelberg

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Vanhaecke, P. (1996). Integrable Hamiltonian systems and symmetric products of curves. In: Integrable Systems in the realm of Algebraic Geometry. Lecture Notes in Mathematics, vol 1638. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21535-7_3

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  • DOI: https://doi.org/10.1007/978-3-662-21535-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61886-7

  • Online ISBN: 978-3-662-21535-7

  • eBook Packages: Springer Book Archive