Poisson Manifolds

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


In 1809 Poisson (see [144]) introduced a bracket on smooth functions, defined on R 2n , by the formula
$$\{ F,G\} : = \sum\limits_{i = 1}^n {\left( {\frac{{\partial F}}{{\partial {q_i}}}\frac{{\partial G}}{{\partial {p_i}}} - \frac{{\partial G}}{{\partial {q_i}}}\frac{{\partial F}}{{\partial {p_i}}}} \right)} .$$
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
$$\begin{array}{*{20}{c}} {{{{\dot{q}}}_{i}} = \{ {{q}_{i}},H\} } \\ {{{{\dot{p}}}_{i}} = \{ {{p}_{i}},H\} } \\ \end{array} i = 1,...,n,$$
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
$$\left\{ {\left\{ {F,G} \right\},H} \right\} + \left\{ {\left\{ {G,H} \right\},F} \right\} + \left\{ {\left\{ {H,F} \right\},G} \right\} = 0$$
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
$$\dot K = \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {q_i}}}} \{ {q_i},H\} + \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {p_i}}}} \{ {p_i},H\} = \{ K,H\} .$$


Poisson Bracket Symplectic Manifold Poisson Structure Jacobi Identity Poisson Manifold 
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© 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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