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The momentum mapping is essentially due to S. Lie, [[#References|[a5]]], pp. 300–343. The modern notion is due to B. Kostant [[#References|[a3]]], J.M. Souriau [[#References|[a9]]] and A.A. Kirillov [[#References|[a2]]].
 
The momentum mapping is essentially due to S. Lie, [[#References|[a5]]], pp. 300–343. The modern notion is due to B. Kostant [[#References|[a3]]], J.M. Souriau [[#References|[a9]]] and A.A. Kirillov [[#References|[a2]]].
  
The setting for the momentum mapping is a smooth [[Symplectic manifold|symplectic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302001.png" /> or even a Poisson manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302002.png" /> (cf. also [[Poisson algebra|Poisson algebra]]; [[Symplectic structure|Symplectic structure]]) with the [[Poisson brackets|Poisson brackets]] on functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302003.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302004.png" /> is the Poisson tensor). To each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302005.png" /> there is the associated Hamiltonian vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302007.png" /> is the [[Lie algebra|Lie algebra]] of all locally Hamiltonian vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302008.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m1302009.png" /> for the [[Lie derivative|Lie derivative]].
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The setting for the momentum mapping is a smooth [[Symplectic manifold|symplectic manifold]] $( M , \omega )$ or even a Poisson manifold $( M , P )$ (cf. also [[Poisson algebra|Poisson algebra]]; [[Symplectic structure|Symplectic structure]]) with the [[Poisson brackets|Poisson brackets]] on functions $\{ f , g \} = P ( d f , d g )$ (where $P = \omega ^ { - 1 } : T ^ { * } M \rightarrow T M$ is the Poisson tensor). To each function $f$ there is the associated Hamiltonian vector field $H _ { f } = P ( d f ) \in \mathfrak{X} ( M , P )$, where $\mathfrak { X } ( M , P )$ is the [[Lie algebra|Lie algebra]] of all locally Hamiltonian vector fields $Y \in \mathfrak { X } ( M )$ satisfying $\mathcal{L} _ { Y } P = 0$ for the [[Lie derivative|Lie derivative]].
  
 
The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie algebra homomorphisms:
 
The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie algebra homomorphisms:
  
<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/m/m130/m130200/m13020010.png" /></td> </tr></table>
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\begin{equation*} 0 \rightarrow H ^ { 0 } ( M ) \rightarrow C ^ { \infty } ( M ) \stackrel { H } { \rightarrow }  \mathfrak{X} ( M , \omega ) \stackrel { \gamma } { \rightarrow } H ^ { 1 } ( M ) \rightarrow 0, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020011.png" />, the [[De Rham cohomology|de Rham cohomology]] class of the contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020012.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020013.png" />, and where the brackets not yet mentioned are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020014.png" />.
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where $\gamma ( Y ) = [ i_{ Y } \omega ]$, the [[De Rham cohomology|de Rham cohomology]] class of the contraction of $Y$ into $\omega$, and where the brackets not yet mentioned are all $0$.
  
A [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020015.png" /> can act from the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020017.png" /> in a way which respects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020018.png" />, so that one obtains a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020020.png" /> is the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020021.png" />. (For a left action one gets an anti-homomorphism of Lie algebras.) One can lift <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020022.png" /> to a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020024.png" />; if not, one replaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020025.png" /> by its Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020026.png" />. The question is whether one can change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020027.png" /> into a homomorphism of Lie algebras. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020028.png" /> then induces a Chevalley <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020030.png" />-cocycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020031.png" />. If it vanishes one can change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020032.png" /> as desired. If not, the cocycle describes a central extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020033.png" /> on which one may change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020034.png" /> to a homomorphism of Lie algebras.
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A [[Lie group|Lie group]] $G$ can act from the right on $M$ by $\alpha : M \times G \rightarrow M$ in a way which respects $\omega$, so that one obtains a homomorphism $\alpha ^ { \prime } : \mathfrak { g } \rightarrow \mathfrak { X } ( M , \omega )$, where $\frak g$ is the Lie algebra of $G$. (For a left action one gets an anti-homomorphism of Lie algebras.) One can lift $\alpha ^ { \prime }$ to a linear mapping $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ if $\gamma \circ \alpha ^ { \prime } = 0$; if not, one replaces $\frak g$ by its Lie subalgebra $\operatorname { ker } ( \gamma \circ \alpha ^ { \prime } ) \subset \mathfrak { g }$. The question is whether one can change $j$ into a homomorphism of Lie algebras. The mapping $\mathfrak { g } \ni X , Y \mapsto \{ j X , j Y \} - j ( [ X , Y ] )$ then induces a Chevalley $2$-cocycle in $H ^ { 2 } ( \mathfrak { g } , H ^ { 0 } ( M ) )$. If it vanishes one can change $j$ as desired. If not, the cocycle describes a central extension of $\frak g$ on which one may change $j$ to a homomorphism of Lie algebras.
  
In any case, even for a Poisson manifold, for a homomorphism of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020035.png" /> (or more generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020036.png" /> is just a linear mapping), by flipping coordinates one gets a momentum mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020037.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020038.png" />-action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020039.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020040.png" /> into the dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020041.png" /> of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020042.png" />,
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In any case, even for a Poisson manifold, for a homomorphism of Lie algebras $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ (or more generally, if $j$ is just a linear mapping), by flipping coordinates one gets a momentum mapping $J$ of the $\frak g$-action $\alpha ^ { \prime }$ from $M$ into the dual $\mathfrak{g} ^ { * }$ of the Lie algebra $\frak g$,
  
<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/m/m130/m130200/m13020043.png" /></td> </tr></table>
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\begin{equation*} J : M \rightarrow \mathfrak { g } ^ { * }, \end{equation*}
  
<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/m/m130/m130200/m13020044.png" /></td> </tr></table>
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\begin{equation*} \langle J ( x ) , X \rangle = j ( X ) ( x ) , H _ { j  ( X ) }= \alpha ^ { \prime } ( X ), \end{equation*}
  
<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/m/m130/m130200/m13020045.png" /></td> </tr></table>
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\begin{equation*} x \in M , X \in \mathfrak { g }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020046.png" /> is the duality pairing.
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where $\langle \, .\, ,\,  . \, \rangle$ is the duality pairing.
  
For a particle in Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020047.png" />-space and the rotation group acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020048.png" />, this is just the angular momentum, hence its name. The momentum mapping is infinitesimally equivariant for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020049.png" />-actions if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020050.png" /> is a homomorphism of Lie algebras. It is a Poisson morphism for the canonical Poisson structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130200/m13020051.png" />, whose symplectic leaves are the co-adjoint orbits. The momentum mapping can be used to reduce the number of coordinates of the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems.
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For a particle in Euclidean $3$-space and the rotation group acting on $T ^ { * } \mathbf{R} ^ { 3 }$, this is just the angular momentum, hence its name. The momentum mapping is infinitesimally equivariant for the $\frak g$-actions if $j$ is a homomorphism of Lie algebras. It is a Poisson morphism for the canonical Poisson structure on $\mathfrak{g} ^ { * }$, whose symplectic leaves are the co-adjoint orbits. The momentum mapping can be used to reduce the number of coordinates of the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems.
  
 
[[#References|[a6]]], [[#References|[a4]]] and [[#References|[a7]]] are convenient references; [[#References|[a7]]] has a large and updated bibliography. The momentum mapping has a strong tendency to have a convex image, and is important for representation theory, see [[#References|[a2]]] and [[#References|[a8]]]. There is also a recent (1998) proposal for a group-valued momentum mapping, see [[#References|[a1]]].
 
[[#References|[a6]]], [[#References|[a4]]] and [[#References|[a7]]] are convenient references; [[#References|[a7]]] has a large and updated bibliography. The momentum mapping has a strong tendency to have a convex image, and is important for representation theory, see [[#References|[a2]]] and [[#References|[a8]]]. There is also a recent (1998) proposal for a group-valued momentum mapping, see [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Alekseev,  A. Malkin,  E. Meinrenken,  "Lie group valued moment maps"  ''J. Diff. Geom.'' , '''48'''  (1998)  pp. 445–495</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Kostant,  "Orbits, symplectic structures, and representation theory" , ''Proc. United States–Japan Sem. Diff. Geom.'' , Nippon Hyoronsha  (1966)  pp. 71</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Libermann,  C.M. Marle,  "Symplectic geometry and analytic mechanics" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lie,  "Theorie der Transformationsgruppen, Zweiter Abschnitt" , Teubner  (1890)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Marmo,  E. Saletan,  A. Simoni,  B. Vitale,  "Dynamical systems. A differential geometric approach to symmetry and reduction" , Wiley/Interscience  (1985)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Marsden,  T. Ratiu,  "Introduction to mechanics and symmetry" , Springer  (1999)  (Edition: Second)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  K.-H. Neeb,  "Holomorphy and convexity in Lie theory" , de Gruyter  (1999)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.M. Souriau,  "Quantification géométrique"  ''Commun. Math. Phys.'' , '''1'''  (1966)  pp. 374–398</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Alekseev,  A. Malkin,  E. Meinrenken,  "Lie group valued moment maps"  ''J. Diff. Geom.'' , '''48'''  (1998)  pp. 445–495</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  B. Kostant,  "Orbits, symplectic structures, and representation theory" , ''Proc. United States–Japan Sem. Diff. Geom.'' , Nippon Hyoronsha  (1966)  pp. 71</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Libermann,  C.M. Marle,  "Symplectic geometry and analytic mechanics" , Reidel  (1987)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S. Lie,  "Theorie der Transformationsgruppen, Zweiter Abschnitt" , Teubner  (1890)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. Marmo,  E. Saletan,  A. Simoni,  B. Vitale,  "Dynamical systems. A differential geometric approach to symmetry and reduction" , Wiley/Interscience  (1985)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J. Marsden,  T. Ratiu,  "Introduction to mechanics and symmetry" , Springer  (1999)  (Edition: Second)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  K.-H. Neeb,  "Holomorphy and convexity in Lie theory" , de Gruyter  (1999)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J.M. Souriau,  "Quantification géométrique"  ''Commun. Math. Phys.'' , '''1'''  (1966)  pp. 374–398</td></tr></table>

Revision as of 16:46, 1 July 2020

The momentum mapping is essentially due to S. Lie, [a5], pp. 300–343. The modern notion is due to B. Kostant [a3], J.M. Souriau [a9] and A.A. Kirillov [a2].

The setting for the momentum mapping is a smooth symplectic manifold $( M , \omega )$ or even a Poisson manifold $( M , P )$ (cf. also Poisson algebra; Symplectic structure) with the Poisson brackets on functions $\{ f , g \} = P ( d f , d g )$ (where $P = \omega ^ { - 1 } : T ^ { * } M \rightarrow T M$ is the Poisson tensor). To each function $f$ there is the associated Hamiltonian vector field $H _ { f } = P ( d f ) \in \mathfrak{X} ( M , P )$, where $\mathfrak { X } ( M , P )$ is the Lie algebra of all locally Hamiltonian vector fields $Y \in \mathfrak { X } ( M )$ satisfying $\mathcal{L} _ { Y } P = 0$ for the Lie derivative.

The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie algebra homomorphisms:

\begin{equation*} 0 \rightarrow H ^ { 0 } ( M ) \rightarrow C ^ { \infty } ( M ) \stackrel { H } { \rightarrow } \mathfrak{X} ( M , \omega ) \stackrel { \gamma } { \rightarrow } H ^ { 1 } ( M ) \rightarrow 0, \end{equation*}

where $\gamma ( Y ) = [ i_{ Y } \omega ]$, the de Rham cohomology class of the contraction of $Y$ into $\omega$, and where the brackets not yet mentioned are all $0$.

A Lie group $G$ can act from the right on $M$ by $\alpha : M \times G \rightarrow M$ in a way which respects $\omega$, so that one obtains a homomorphism $\alpha ^ { \prime } : \mathfrak { g } \rightarrow \mathfrak { X } ( M , \omega )$, where $\frak g$ is the Lie algebra of $G$. (For a left action one gets an anti-homomorphism of Lie algebras.) One can lift $\alpha ^ { \prime }$ to a linear mapping $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ if $\gamma \circ \alpha ^ { \prime } = 0$; if not, one replaces $\frak g$ by its Lie subalgebra $\operatorname { ker } ( \gamma \circ \alpha ^ { \prime } ) \subset \mathfrak { g }$. The question is whether one can change $j$ into a homomorphism of Lie algebras. The mapping $\mathfrak { g } \ni X , Y \mapsto \{ j X , j Y \} - j ( [ X , Y ] )$ then induces a Chevalley $2$-cocycle in $H ^ { 2 } ( \mathfrak { g } , H ^ { 0 } ( M ) )$. If it vanishes one can change $j$ as desired. If not, the cocycle describes a central extension of $\frak g$ on which one may change $j$ to a homomorphism of Lie algebras.

In any case, even for a Poisson manifold, for a homomorphism of Lie algebras $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ (or more generally, if $j$ is just a linear mapping), by flipping coordinates one gets a momentum mapping $J$ of the $\frak g$-action $\alpha ^ { \prime }$ from $M$ into the dual $\mathfrak{g} ^ { * }$ of the Lie algebra $\frak g$,

\begin{equation*} J : M \rightarrow \mathfrak { g } ^ { * }, \end{equation*}

\begin{equation*} \langle J ( x ) , X \rangle = j ( X ) ( x ) , H _ { j ( X ) }= \alpha ^ { \prime } ( X ), \end{equation*}

\begin{equation*} x \in M , X \in \mathfrak { g }, \end{equation*}

where $\langle \, .\, ,\, . \, \rangle$ is the duality pairing.

For a particle in Euclidean $3$-space and the rotation group acting on $T ^ { * } \mathbf{R} ^ { 3 }$, this is just the angular momentum, hence its name. The momentum mapping is infinitesimally equivariant for the $\frak g$-actions if $j$ is a homomorphism of Lie algebras. It is a Poisson morphism for the canonical Poisson structure on $\mathfrak{g} ^ { * }$, whose symplectic leaves are the co-adjoint orbits. The momentum mapping can be used to reduce the number of coordinates of the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems.

[a6], [a4] and [a7] are convenient references; [a7] has a large and updated bibliography. The momentum mapping has a strong tendency to have a convex image, and is important for representation theory, see [a2] and [a8]. There is also a recent (1998) proposal for a group-valued momentum mapping, see [a1].

References

[a1] A. Alekseev, A. Malkin, E. Meinrenken, "Lie group valued moment maps" J. Diff. Geom. , 48 (1998) pp. 445–495
[a2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976)
[a3] B. Kostant, "Orbits, symplectic structures, and representation theory" , Proc. United States–Japan Sem. Diff. Geom. , Nippon Hyoronsha (1966) pp. 71
[a4] P. Libermann, C.M. Marle, "Symplectic geometry and analytic mechanics" , Reidel (1987)
[a5] S. Lie, "Theorie der Transformationsgruppen, Zweiter Abschnitt" , Teubner (1890)
[a6] G. Marmo, E. Saletan, A. Simoni, B. Vitale, "Dynamical systems. A differential geometric approach to symmetry and reduction" , Wiley/Interscience (1985)
[a7] J. Marsden, T. Ratiu, "Introduction to mechanics and symmetry" , Springer (1999) (Edition: Second)
[a8] K.-H. Neeb, "Holomorphy and convexity in Lie theory" , de Gruyter (1999)
[a9] J.M. Souriau, "Quantification géométrique" Commun. Math. Phys. , 1 (1966) pp. 374–398
How to Cite This Entry:
Momentum mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Momentum_mapping&oldid=50026
This article was adapted from an original article by Peter W. Michor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article