Skip to main content

Integrable Hamiltonian systems on affine Poisson varieties

  • Chapter
  • 641 Accesses

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

Abstract

In this chapter we give the basic definitions and properties of integrable Hamiltonian systems on affine Poisson varieties and their morphisms. In Section 2 we define the notion of a Poisson bracket (or Poisson structure) on an affine algebraic variety (over C). The Poisson bracket is precisely what is needed to define Hamiltonian mechanics on a space, as is well-known from the theory of symplectic and Poisson manifolds. We shortly describe the simplest Poisson structures (i.e., linear, affine and quadratic structures; also structures on C2 and C3) and describe two natural decompositions of affine Poisson varieties, one is given by the algebra of Casimirs, the other comes from the notion of rank of a Poisson structure (at a point). We also describe several ways to build new affine Poisson varieties from old ones.

Keywords

  • Poisson Bracket
  • Poisson Structure
  • Poisson Algebra
  • Integrable Hamiltonian System
  • Poisson Manifold

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Vanhaecke, P. (1996). Integrable Hamiltonian systems on affine Poisson varieties. 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_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21535-7_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61886-7

  • Online ISBN: 978-3-662-21535-7

  • eBook Packages: Springer Book Archive