Poisson manifold

In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a field in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Poisson manifold is a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> endowed with a Poisson structure. The notion of Poisson manifold generalises that of <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifold</a>, which in turn generalises the <a href="/facts/Phase_space/qPe4Fq57">phase space</a> from <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a>.
A Poisson structure (or Poisson bracket) on a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a function
 
 
 
 {
 ⋅
 ,
 ⋅
 }
 :
 
 
 
 C
 
 
 
 ∞
 
 
 (
 M
 )
 ×
 
 
 
 C
 
 
 
 ∞
 
 
 (
 M
 )
 →
 
 
 
 C
 
 
 
 ∞
 
 
 (
 M
 )
 
 
 {\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}
 
on the <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 
 
 
 C
 
 
 
 ∞
 
 
 (
 M
 )
 
 
 {\displaystyle {\mathcal {C}}^{\infty }(M)}
 
 of <a href="/facts/Smooth_functions/mffL4ch2">smooth functions</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
, making it into a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> subject to a <a href="/facts/Product_rule/MyKxsP5t">Leibniz rule</a> (also known as a <a href="/facts/Poisson_algebra/dRkCcexQ">Poisson algebra</a>).
Poisson structures on manifolds were introduced by <a href="/facts/Andr%25C3%25A9_Lichnerowicz/4AlCVKAP">André Lichnerowicz</a> in 1977 and are named after the French mathematician <a href="/facts/Sim%25C3%25A9on_Denis_Poisson/zmsDITrY">Siméon Denis Poisson</a>, due to their early appearance in his works on <a href="/facts/Analytical_mechanics/aotB0n4t">analytical mechanics</a>.

Poisson manifold open-in-new

Poisson manifold