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Poisson Manifolds

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 47)

Abstract

In 1809 Poisson (see [144]) introduced a bracket on smooth functions, defined on R 2n , by the formula
$$\{ F,G\} : = \sum\limits_{i = 1}^n {\left( {\frac{{\partial F}}{{\partial {q_i}}}\frac{{\partial G}}{{\partial {p_i}}} - \frac{{\partial G}}{{\partial {q_i}}}\frac{{\partial F}}{{\partial {p_i}}}} \right)} .$$
(3.1)
In this formula, F and G are arbitrary smooth functions on R 2n and (q 1, ..., q n , p 1 ..., p n ) are linear coordinates on R 2n . He observed that if F and G are two first integrals of a mechanical system (defined on R 2n ) then their Poisson bracket {F, G} is also a first integral. Notice that the Poisson bracket also allows one to describe the equations of motion in their most symmetric form
$$\begin{array}{*{20}{c}} {{{{\dot{q}}}_{i}} = \{ {{q}_{i}},H\} } \\ {{{{\dot{p}}}_{i}} = \{ {{p}_{i}},H\} } \\ \end{array} i = 1,...,n,$$
(3.2)
where H: R 2n R is the Hamiltonian (the energy of the mechanical system, expressed in terms of position and momentum). Thirty years later, Jacobi explained (in [90]) Poisson’s observation by showing that the bracket (3.1) satisfies the identity
$$\left\{ {\left\{ {F,G} \right\},H} \right\} + \left\{ {\left\{ {G,H} \right\},F} \right\} + \left\{ {\left\{ {H,F} \right\},G} \right\} = 0$$
(3.3)
for all smooth functions F, G, H defined on R 2n . The above identity is now known as the Jacobi identity. To see how Poisson’s Theorem follows from the Jacobi identity it suffices to remark that K is a constant of the motion (3.2), precisely if K Poisson-commutes with H, i.e., {K, H} = 0, since
$$\dot K = \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {q_i}}}} \{ {q_i},H\} + \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {p_i}}}} \{ {p_i},H\} = \{ K,H\} .$$

Keywords

Poisson Bracket Symplectic Manifold Poisson Structure Jacobi Identity 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.Department of MathematicsUniversity of LouvainLouvain-la-NeuveBelgium
  3. 3.Laboratoire de Mathématiques et ApplicationsUniversité de PoitiersFuturoscopeFrance

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