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The differential expression
 
The differential expression
  
<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/p/p073/p073270/p0732701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
( u , v )  = \
 +
\sum _ { i= } 1 ^ { n }
 +
\left (
 +
 
 +
\frac{\partial  u }{\partial  q _ {i} }
 +
 
 +
\frac{\partial  v }{\partial  p _ {i} }
 +
-
 +
 
 +
\frac{\partial  u }{\partial  p _ {i} }
 +
 
 +
\frac{\partial  v }{\partial  q _ {i} }
 +
 
 +
\right ) ,
 +
$$
  
depending on two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732703.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732704.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732706.png" />. The Poisson brackets, introduced by S. Poisson [[#References|[1]]], are a particular case of the [[Jacobi brackets|Jacobi brackets]]. The Poisson brackets are a bilinear form in the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p0732708.png" />, such that
+
depending on two functions $  u ( q , p ) $
 +
and $  v ( q , p ) $
 +
of $  2n $
 +
variables  $  q = ( q _ {1} \dots q _ {n} ) $,
 +
p = ( p _ {1} \dots p _ {n} ) $.  
 +
The Poisson brackets, introduced by S. Poisson [[#References|[1]]], are a particular case of the [[Jacobi brackets|Jacobi brackets]]. The Poisson brackets are a bilinear form in the functions $  u $
 +
and $  v $,  
 +
such that
  
<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/p/p073/p073270/p0732709.png" /></td> </tr></table>
+
$$
 +
( u , v )  = - ( v , u )
 +
$$
  
 
and the Jacobi identity
 
and the Jacobi identity
  
<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/p/p073/p073270/p07327010.png" /></td> </tr></table>
+
$$
 +
( u , ( v , w ) ) +
 +
( v , ( w , u ) ) +
 +
( w , ( u , v ) )  = 0
 +
$$
  
 
holds (see [[#References|[2]]]).
 
holds (see [[#References|[2]]]).
  
The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see [[#References|[3]]]–[[#References|[5]]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327012.png" /> are canonical variables and a transformation
+
The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see [[#References|[3]]]–[[#References|[5]]]). For example, if $  q $
 +
and p $
 +
are canonical variables and a transformation
  
<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/p/p073/p073270/p07327013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= Q ( q , p ) ,\ \
 +
= P ( q , p )
 +
$$
  
is given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327015.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327016.png" />-matrices
+
is given, where $  Q = ( Q _ {1} \dots Q _ {n} ) $,  
 +
$  P = ( P _ {1} \dots P _ {n} ) $
 +
and the $  ( n \times n ) $-
 +
matrices
  
<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/p/p073/p073270/p07327017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
( P , P ) ,\  ( Q , Q ) ,\  ( Q , P )
 +
$$
  
are constructed with entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327020.png" />, respectively, then (2) is a canonical transformation if and only if the first two matrices in (3) are zero and the third is the unit matrix.
+
are constructed with entries $  ( P _ {i} , P _ {j} ) $,  
 +
$  ( Q _ {i} , Q _ {j} ) $,
 +
$  ( Q _ {i} , P _ {j} ) $,
 +
respectively, then (2) is a canonical transformation if and only if the first two matrices in (3) are zero and the third is the unit matrix.
  
The Poisson brackets, computed for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327022.png" /> are replaced in (1) by some pair of coordinate functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327024.png" />, are also called fundamental brackets.
+
The Poisson brackets, computed for the case when $  u $
 +
and $  v $
 +
are replaced in (1) by some pair of coordinate functions in $  q $
 +
and p $,  
 +
are also called fundamental brackets.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Poisson,  ''J. Ecole Polytechn.'' , '''8'''  (1809)  pp. 266–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G.J. Jacobi,  "Nova methodus, aequationes differentiales partiales primi ordinis inter numurum variabilium quemcunque propositas integrandi"  ''J. Reine Angew. Math.'' , '''60'''  (1862)  pp. 1–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Lur'e,  "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Poisson,  ''J. Ecole Polytechn.'' , '''8'''  (1809)  pp. 266–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G.J. Jacobi,  "Nova methodus, aequationes differentiales partiales primi ordinis inter numurum variabilium quemcunque propositas integrandi"  ''J. Reine Angew. Math.'' , '''60'''  (1862)  pp. 1–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Lur'e,  "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>
  
 +
====Comments====
 +
Other basic properties of Poisson brackets are invariance under canonical transformations and the fact that  $  ( F, H) $
 +
expresses the derivative of  $  F( q, p) $
 +
along trajectories, if  $  H $
 +
is the [[Hamiltonian|Hamiltonian]], so that the corresponding Hamiltonian equations are  $  \dot{q} _ {i} = ( q _ {i} , H) $,
 +
$  \dot{p} _ {i} =( p _ {i} , H) $,
 +
which for a  "standard"  Hamiltonian of the form  $  H=( \sum p _ {i}  ^ {2} )/2+ V( q) $
 +
gives back  $  \dot{q} _ {i} = p _ {i} $,
 +
$  \dot{p} _ {i} = - \partial  H/ \partial  q _ {i} $.
 +
Therefore  $  ( F, H) $
 +
expresses a conservation law, i.e.  $  F $
 +
is a conserved quantity.
 +
 +
The Poisson brackets may be defined for functionals depending on a function  $  q( x) $,
 +
as
  
 +
$$
 +
F[ q]  =  \int\limits _ {- \infty } ^  \infty  \widetilde{F}  ( q  ,q  ^ {(} 1) , q  ^ {(} 2) ,\dots)  dx,
 +
$$
  
====Comments====
+
with  $ q ^ {(} n) = d  ^ {n} q/dx  ^ {n} $.
Other basic properties of Poisson brackets are invariance under canonical transformations and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327025.png" /> expresses the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327026.png" /> along trajectories, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327027.png" /> is the [[Hamiltonian|Hamiltonian]], so that the corresponding Hamiltonian equations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327029.png" />, which for a "standard" Hamiltonian of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327030.png" /> gives back <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327032.png" />. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327033.png" /> expresses a conservation law, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327034.png" /> is a conserved quantity.
 
  
The Poisson brackets may be defined for functionals depending on a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327035.png" />, as
+
One has
  
<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/p/p073/p073270/p07327036.png" /></td> </tr></table>
+
$$
 +
( F, G)  = \int\limits _ {- \infty } ^  \infty 
 +
\frac{\delta \widetilde{F}  }{\delta q }
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327037.png" />.
+
\frac{d}{dx}
 +
 +
\frac{\delta \widetilde{G}  }{\delta q }
 +
  dx,
 +
$$
  
One has
+
with  $  {\delta \widetilde{F}  } / {\delta q } $,
 +
$  {\delta \widetilde{G}  } / {\delta q } $
 +
variational derivatives, i.e.
  
<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/p/p073/p073270/p07327038.png" /></td> </tr></table>
+
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073270/p07327040.png" /> variational derivatives, i.e.
+
\frac{\delta \widetilde{F}  }{\delta q }
 +
  = \sum \left ( -
 +
\frac{d}{dx}
 +
\right )  ^ {n}
  
<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/p/p073/p073270/p07327041.png" /></td> </tr></table>
+
\frac{\partial  \widetilde{F}  }{\partial  q  ^ {(} n) }
 +
.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Newell,  "Solitons in mathematical physics" , SIAM  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Newell,  "Solitons in mathematical physics" , SIAM  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>

Revision as of 08:06, 6 June 2020


The differential expression

$$ \tag{1 } ( u , v ) = \ \sum _ { i= } 1 ^ { n } \left ( \frac{\partial u }{\partial q _ {i} } \frac{\partial v }{\partial p _ {i} } - \frac{\partial u }{\partial p _ {i} } \frac{\partial v }{\partial q _ {i} } \right ) , $$

depending on two functions $ u ( q , p ) $ and $ v ( q , p ) $ of $ 2n $ variables $ q = ( q _ {1} \dots q _ {n} ) $, $ p = ( p _ {1} \dots p _ {n} ) $. The Poisson brackets, introduced by S. Poisson [1], are a particular case of the Jacobi brackets. The Poisson brackets are a bilinear form in the functions $ u $ and $ v $, such that

$$ ( u , v ) = - ( v , u ) $$

and the Jacobi identity

$$ ( u , ( v , w ) ) + ( v , ( w , u ) ) + ( w , ( u , v ) ) = 0 $$

holds (see [2]).

The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see [3][5]). For example, if $ q $ and $ p $ are canonical variables and a transformation

$$ \tag{2 } Q = Q ( q , p ) ,\ \ P = P ( q , p ) $$

is given, where $ Q = ( Q _ {1} \dots Q _ {n} ) $, $ P = ( P _ {1} \dots P _ {n} ) $ and the $ ( n \times n ) $- matrices

$$ \tag{3 } ( P , P ) ,\ ( Q , Q ) ,\ ( Q , P ) $$

are constructed with entries $ ( P _ {i} , P _ {j} ) $, $ ( Q _ {i} , Q _ {j} ) $, $ ( Q _ {i} , P _ {j} ) $, respectively, then (2) is a canonical transformation if and only if the first two matrices in (3) are zero and the third is the unit matrix.

The Poisson brackets, computed for the case when $ u $ and $ v $ are replaced in (1) by some pair of coordinate functions in $ q $ and $ p $, are also called fundamental brackets.

References

[1] S. Poisson, J. Ecole Polytechn. , 8 (1809) pp. 266–344
[2] C.G.J. Jacobi, "Nova methodus, aequationes differentiales partiales primi ordinis inter numurum variabilium quemcunque propositas integrandi" J. Reine Angew. Math. , 60 (1862) pp. 1–181
[3] E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[4] A.I. Lur'e, "Analytical mechanics" , Moscow (1961) (In Russian)
[5] H. Goldstein, "Classical mechanics" , Addison-Wesley (1957)

Comments

Other basic properties of Poisson brackets are invariance under canonical transformations and the fact that $ ( F, H) $ expresses the derivative of $ F( q, p) $ along trajectories, if $ H $ is the Hamiltonian, so that the corresponding Hamiltonian equations are $ \dot{q} _ {i} = ( q _ {i} , H) $, $ \dot{p} _ {i} =( p _ {i} , H) $, which for a "standard" Hamiltonian of the form $ H=( \sum p _ {i} ^ {2} )/2+ V( q) $ gives back $ \dot{q} _ {i} = p _ {i} $, $ \dot{p} _ {i} = - \partial H/ \partial q _ {i} $. Therefore $ ( F, H) $ expresses a conservation law, i.e. $ F $ is a conserved quantity.

The Poisson brackets may be defined for functionals depending on a function $ q( x) $, as

$$ F[ q] = \int\limits _ {- \infty } ^ \infty \widetilde{F} ( q ,q ^ {(} 1) , q ^ {(} 2) ,\dots) dx, $$

with $ q ^ {(} n) = d ^ {n} q/dx ^ {n} $.

One has

$$ ( F, G) = \int\limits _ {- \infty } ^ \infty \frac{\delta \widetilde{F} }{\delta q } \frac{d}{dx} \frac{\delta \widetilde{G} }{\delta q } dx, $$

with $ {\delta \widetilde{F} } / {\delta q } $, $ {\delta \widetilde{G} } / {\delta q } $ variational derivatives, i.e.

$$ \frac{\delta \widetilde{F} }{\delta q } = \sum \left ( - \frac{d}{dx} \right ) ^ {n} \frac{\partial \widetilde{F} }{\partial q ^ {(} n) } . $$

References

[a1] A.C. Newell, "Solitons in mathematical physics" , SIAM (1985)
[a2] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a3] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978)
[a4] F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)
How to Cite This Entry:
Poisson brackets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_brackets&oldid=48215
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article