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An [[Infinitesimal structure|infinitesimal structure]] of order one on an even-dimensional smooth orientable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918601.png" /> which is defined by a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918602.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918603.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918604.png" />. Every tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918605.png" /> has the structure of a symplectic space with skew-symmetric scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918606.png" />. All frames tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918607.png" /> adapted to the symplectic structure (that is, frames with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918608.png" /> has the canonical form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918609.png" />) form a principal fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186010.png" /> whose structure group is the [[Symplectic group|symplectic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186011.png" />. Specifying a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186012.png" /> is equivalent to specifying an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186013.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186014.png" /> (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186015.png" />-structure]]).
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Given a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186016.png" />, there is an isomorphism between the modules of vector fields and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186017.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186018.png" />, under which a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186019.png" /> is associated with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186020.png" />-form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186021.png" />. In this context, the image of the Lie bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186022.png" /> is called the Poisson bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186023.png" />. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186025.png" /> are exact differentials, one obtains the concept of the Poisson bracket of two functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186026.png" />, which generalizes the corresponding classical concept.
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A symplectic structure is also called an almost-Hamiltonian structure, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186027.png" /> is closed, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186028.png" />, a Hamiltonian structure, though the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186029.png" /> is sometimes included in the definition of a symplectic structure. These structures find application in global analytical mechanics, since the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186030.png" /> of any smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186031.png" /> admits a canonical Hamiltonian structure. It is defined by the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186032.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186033.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186035.png" />, called the Liouville form, is given by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186036.png" /> for any tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186037.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186039.png" /> is the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186040.png" />. If one chooses local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186044.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186045.png" />. In classical mechanics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186046.png" /> is interpreted as the configuration space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186047.png" /> as the phase space.
+
An [[Infinitesimal structure|infinitesimal structure]] of order one on an even-dimensional smooth orientable manifold  $  M  ^ {2n} $
 +
which is defined by a non-degenerate  $  2 $-
 +
form  $  \Phi $
 +
on  $  M  ^ {2n} $.  
 +
Every tangent space  $  T _ {x} ( M  ^ {2n} ) $
 +
has the structure of a symplectic space with skew-symmetric scalar product  $  \Phi ( X, Y) $.  
 +
All frames tangent to  $  M  ^ {2n} $
 +
adapted to the symplectic structure (that is, frames with respect to which  $  \Phi $
 +
has the canonical form $  \Phi = 2 \sum _ {\alpha = 1 }  ^ {n} \omega  ^  \alpha  \wedge \omega ^ {n + \alpha } $)
 +
form a principal fibre bundle over  $  M  ^ {2n} $
 +
whose structure group is the [[Symplectic group|symplectic group]]  $  \mathop{\rm Sp} ( n) $.  
 +
Specifying a symplectic structure on  $  M  ^ {2n} $
 +
is equivalent to specifying an  $  \mathop{\rm Sp} ( n) $-
 +
structure on $  M  ^ {2n} $(
 +
cf. [[G-structure| $  G $-
 +
structure]]).
  
A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186048.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186049.png" /> with a Hamiltonian structure is called a Hamiltonian vector field (or a Hamiltonian system) if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186050.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186051.png" /> is closed. If, in addition, it is exact, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186053.png" /> is called a Hamiltonian on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186054.png" /> and is a generalization of the corresponding classical concept.
+
Given a symplectic structure on  $  M  ^ {2n} $,
 +
there is an isomorphism between the modules of vector fields and  $  1 $-
 +
forms on  $  M  ^ {2n} $,
 +
under which a vector field $  X $
 +
is associated with a  $  1 $-
 +
form,  $  \omega _ {X} :  Y \mapsto \Phi ( X, Y) $.
 +
In this context, the image of the Lie bracket  $  [ X, Y] $
 +
is called the Poisson bracket  $  [ \omega _ {X} , \omega _ {Y} ] $.
 +
In particular, when  $  \omega _ {X} $
 +
and  $  \omega _ {Y} $
 +
are exact differentials, one obtains the concept of the Poisson bracket of two functions on  $  M  ^ {2n} $,
 +
which generalizes the corresponding classical concept.
 +
 
 +
A symplectic structure is also called an almost-Hamiltonian structure, and if  $  \Phi $
 +
is closed, i.e.  $  d \Phi = 0 $,
 +
a Hamiltonian structure, though the condition  $  d \Phi = 0 $
 +
is sometimes included in the definition of a symplectic structure. These structures find application in global analytical mechanics, since the cotangent bundle  $  T  ^ {*} ( M) $
 +
of any smooth manifold  $  M $
 +
admits a canonical Hamiltonian structure. It is defined by the form  $  \Phi = d \theta $,
 +
where the  $  1 $-
 +
form  $  \theta $
 +
on  $  T  ^ {*} ( M) $,
 +
called the Liouville form, is given by: $  \theta _ {u} ( X _ {u} ) = u ( \pi _ {*} X _ {u} ) $
 +
for any tangent vector  $  X _ {u} $
 +
at the point  $  u \in T  ^ {*} ( M) $,
 +
where  $  \pi $
 +
is the projection  $  T  ^ {*} ( M) \rightarrow M $.  
 +
If one chooses local coordinates  $  x  ^ {1} \dots x  ^ {n} $
 +
on $  M $,
 +
and  $  u = y _ {i} ( u)  dx _ {\pi ( u) }  ^ {i} $,
 +
then  $  \theta = y _ {i}  dx  ^ {i} $,
 +
so that  $  \Phi = dy _ {i} \wedge dx  ^ {i} $.  
 +
In classical mechanics  $  M $
 +
is interpreted as the configuration space and  $  T  ^ {*} ( M) $
 +
as the phase space.
 +
 
 +
A vector field  $  X $
 +
on a manifold  $  M  ^ {2n} $
 +
with a Hamiltonian structure is called a Hamiltonian vector field (or a Hamiltonian system) if the $  1 $-
 +
form $  \omega _ {X} $
 +
is closed. If, in addition, it is exact, that is, $  \omega _ {X} = - dH $,  
 +
then $  H $
 +
is called a Hamiltonian on $  M  ^ {2n} $
 +
and is a generalization of the corresponding classical concept.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Godbillon,  "Géométrie différentielle et mécanique analytique" , Hermann  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Godbillon,  "Géométrie différentielle et mécanique analytique" , Hermann  (1969)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Mostly, for a symplectic structure on a manifold the defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186055.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186056.png" /> is required to be closed (cf. [[#References|[a1]]], p. 176, [[#References|[a4]]], p. 36ff). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186057.png" /> is not necessarily closed, one speaks of an almost-symplectic structure.
+
Mostly, for a symplectic structure on a manifold the defining $  2 $-
 +
form $  \Phi $
 +
is required to be closed (cf. [[#References|[a1]]], p. 176, [[#References|[a4]]], p. 36ff). If $  \Phi $
 +
is not necessarily closed, one speaks of an almost-symplectic structure.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186058.png" /> denote the vector field on a symplectic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186059.png" /> that corresponds to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186060.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186061.png" />. Then the Poisson bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186062.png" /> is defined by
+
Let $  \Phi ( \omega ) $
 +
denote the vector field on a symplectic manifold $  M $
 +
that corresponds to the $  1 $-
 +
form $  \omega $.  
 +
Then the Poisson bracket on $  C  ^  \infty  ( M) $
 +
is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186063.png" /></td> </tr></table>
+
$$
 +
\{ f, g \}  = \Phi ( \phi ( df), \phi ( dg)) .
 +
$$
  
This turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186064.png" /> into a Lie algebra which satisfies the Leibniz property
+
This turns $  C  ^  \infty  ( M) $
 +
into a Lie algebra which satisfies the Leibniz property
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\{ f, gh \}  = \{ f, g \} h + g \{ f, h \} .
 +
$$
  
More generally, an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186066.png" /> which has an extra Lie bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186067.png" /> so that (*) is satisfied is called a Poisson algebra. A smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186068.png" /> with a Poisson algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186069.png" /> is called a Poisson manifold, [[#References|[a4]]], p. 107ff.
+
More generally, an algebra $  A $
 +
which has an extra Lie bracket $  \{  , \} $
 +
so that (*) is satisfied is called a Poisson algebra. A smooth manifold $  M $
 +
with a Poisson algebra structure on $  C  ^  \infty  ( M) $
 +
is called a Poisson manifold, [[#References|[a4]]], p. 107ff.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Souriau,  "Structures des systèmes dynamiques" , Dunod  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.I. [V.I. Arnol'd] Arnold,  A.B. [A.B. Givent'al] Giventhal,  "Symplectic geometry" , ''Dynamical Systems'' , '''IV''' , Springer  (1990)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Crumeyrolle (ed.)  J Grifone (ed.) , ''Symplectic geometry'' , Pitman  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Souriau,  "Structures des systèmes dynamiques" , Dunod  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.I. [V.I. Arnol'd] Arnold,  A.B. [A.B. Givent'al] Giventhal,  "Symplectic geometry" , ''Dynamical Systems'' , '''IV''' , Springer  (1990)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Crumeyrolle (ed.)  J Grifone (ed.) , ''Symplectic geometry'' , Pitman  (1983)</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


An infinitesimal structure of order one on an even-dimensional smooth orientable manifold $ M ^ {2n} $ which is defined by a non-degenerate $ 2 $- form $ \Phi $ on $ M ^ {2n} $. Every tangent space $ T _ {x} ( M ^ {2n} ) $ has the structure of a symplectic space with skew-symmetric scalar product $ \Phi ( X, Y) $. All frames tangent to $ M ^ {2n} $ adapted to the symplectic structure (that is, frames with respect to which $ \Phi $ has the canonical form $ \Phi = 2 \sum _ {\alpha = 1 } ^ {n} \omega ^ \alpha \wedge \omega ^ {n + \alpha } $) form a principal fibre bundle over $ M ^ {2n} $ whose structure group is the symplectic group $ \mathop{\rm Sp} ( n) $. Specifying a symplectic structure on $ M ^ {2n} $ is equivalent to specifying an $ \mathop{\rm Sp} ( n) $- structure on $ M ^ {2n} $( cf. $ G $- structure).

Given a symplectic structure on $ M ^ {2n} $, there is an isomorphism between the modules of vector fields and $ 1 $- forms on $ M ^ {2n} $, under which a vector field $ X $ is associated with a $ 1 $- form, $ \omega _ {X} : Y \mapsto \Phi ( X, Y) $. In this context, the image of the Lie bracket $ [ X, Y] $ is called the Poisson bracket $ [ \omega _ {X} , \omega _ {Y} ] $. In particular, when $ \omega _ {X} $ and $ \omega _ {Y} $ are exact differentials, one obtains the concept of the Poisson bracket of two functions on $ M ^ {2n} $, which generalizes the corresponding classical concept.

A symplectic structure is also called an almost-Hamiltonian structure, and if $ \Phi $ is closed, i.e. $ d \Phi = 0 $, a Hamiltonian structure, though the condition $ d \Phi = 0 $ is sometimes included in the definition of a symplectic structure. These structures find application in global analytical mechanics, since the cotangent bundle $ T ^ {*} ( M) $ of any smooth manifold $ M $ admits a canonical Hamiltonian structure. It is defined by the form $ \Phi = d \theta $, where the $ 1 $- form $ \theta $ on $ T ^ {*} ( M) $, called the Liouville form, is given by: $ \theta _ {u} ( X _ {u} ) = u ( \pi _ {*} X _ {u} ) $ for any tangent vector $ X _ {u} $ at the point $ u \in T ^ {*} ( M) $, where $ \pi $ is the projection $ T ^ {*} ( M) \rightarrow M $. If one chooses local coordinates $ x ^ {1} \dots x ^ {n} $ on $ M $, and $ u = y _ {i} ( u) dx _ {\pi ( u) } ^ {i} $, then $ \theta = y _ {i} dx ^ {i} $, so that $ \Phi = dy _ {i} \wedge dx ^ {i} $. In classical mechanics $ M $ is interpreted as the configuration space and $ T ^ {*} ( M) $ as the phase space.

A vector field $ X $ on a manifold $ M ^ {2n} $ with a Hamiltonian structure is called a Hamiltonian vector field (or a Hamiltonian system) if the $ 1 $- form $ \omega _ {X} $ is closed. If, in addition, it is exact, that is, $ \omega _ {X} = - dH $, then $ H $ is called a Hamiltonian on $ M ^ {2n} $ and is a generalization of the corresponding classical concept.

References

[1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[2] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)

Comments

Mostly, for a symplectic structure on a manifold the defining $ 2 $- form $ \Phi $ is required to be closed (cf. [a1], p. 176, [a4], p. 36ff). If $ \Phi $ is not necessarily closed, one speaks of an almost-symplectic structure.

Let $ \Phi ( \omega ) $ denote the vector field on a symplectic manifold $ M $ that corresponds to the $ 1 $- form $ \omega $. Then the Poisson bracket on $ C ^ \infty ( M) $ is defined by

$$ \{ f, g \} = \Phi ( \phi ( df), \phi ( dg)) . $$

This turns $ C ^ \infty ( M) $ into a Lie algebra which satisfies the Leibniz property

$$ \tag{* } \{ f, gh \} = \{ f, g \} h + g \{ f, h \} . $$

More generally, an algebra $ A $ which has an extra Lie bracket $ \{ , \} $ so that (*) is satisfied is called a Poisson algebra. A smooth manifold $ M $ with a Poisson algebra structure on $ C ^ \infty ( M) $ is called a Poisson manifold, [a4], p. 107ff.

References

[a1] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)
[a2] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a3] J.M. Souriau, "Structures des systèmes dynamiques" , Dunod (1969)
[a4] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)
[a5] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a6] V.I. [V.I. Arnol'd] Arnold, A.B. [A.B. Givent'al] Giventhal, "Symplectic geometry" , Dynamical Systems , IV , Springer (1990) (Translated from Russian)
[a7] A. Crumeyrolle (ed.) J Grifone (ed.) , Symplectic geometry , Pitman (1983)
How to Cite This Entry:
Symplectic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_structure&oldid=14696
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article