# Momentum map

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Momentum_map
> Markdown URL: https://mediated.wiki/source/Momentum_map.md
> Source: https://en.wikipedia.org/wiki/Momentum_map
> Source revision: 1356257378
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Tool in symplectic geometry}}
In [mathematics](/source/mathematics), specifically in [symplectic geometry](/source/symplectic_geometry), the '''momentum map''' (or, by false etymology, '''moment map'''<ref>''Moment map'' is a misnomer and physically incorrect. It is an erroneous translation of the French notion ''application moment''. See [https://mathoverflow.net/q/242468 this mathoverflow question] for the history of the name.</ref>) is a tool associated with a [Hamiltonian](/source/Hamiltonian_action) [action](/source/Group_action_(mathematics)) of a [Lie group](/source/Lie_group) on a [symplectic manifold](/source/symplectic_manifold), used to construct [conserved quantities](/source/conserved_quantities) for the action. The momentum map generalizes the classical notions of linear and angular [momentum](/source/momentum). It is an essential ingredient in various constructions of symplectic manifolds, including '''symplectic''' ('''Marsden–Weinstein''') '''quotients''', discussed below, and [symplectic cut](/source/symplectic_cut)s and [sums](/source/symplectic_sum).

== Formal definition ==
Let <math>M</math> be a manifold with [symplectic form](/source/Symplectic_manifold) ''<math>\omega</math>''. Suppose that a Lie group ''<math>G</math>'' acts on ''<math>M</math>'' via [symplectomorphism](/source/symplectomorphism)s (that is, the action of each ''<math>g</math>'' in ''<math>G</math>'' preserves ''<math>\omega</math>''). Let <math>\mathfrak{g}</math> be the [Lie algebra](/source/Lie_algebra) of ''<math>G</math>'', <math>\mathfrak{g}^*</math> its [dual](/source/dual_space), and

:<math>\langle \, \cdot, \cdot\rangle : \mathfrak{g}^* \times \mathfrak{g} \to \mathbb{R}</math>

the pairing between the two. Any ''<math>\xi</math>'' in <math>\mathfrak{g}</math> induces a [vector field](/source/vector_field) ''<math>\rho(\xi)</math>'' on ''<math>M</math>'' describing the [infinitesimal](/source/infinitesimal) action of ''<math>\xi</math>''. To be precise, at a point ''<math>x</math>'' in ''<math>M</math>'' the vector <math>\rho(\xi)_x</math> is

:<math>\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t = 0} \exp(t \xi) \cdot x,</math>

where <math>\exp : \mathfrak{g} \to G</math> is the [exponential map](/source/exponential_map_(Lie_theory)) and <math>\cdot</math> denotes the ''<math>G</math>''-action on ''<math>M</math>''.<ref>The vector field ρ(ξ) is called sometimes the [Killing vector field](/source/Killing_vector_field) relative to the action of the [one-parameter subgroup](/source/Exponential_map_(Lie_theory)) generated by ξ. See, for instance, {{harv|Choquet-Bruhat|DeWitt-Morette|1977}}</ref> Let <math>\iota_{\rho(\xi)} \omega \,</math> denote the [contraction](/source/Interior_product) of this vector field with ''<math>\omega</math>''. Because ''<math>G</math>'' acts by symplectomorphisms, it follows that <math>\iota_{\rho(\xi)} \omega \,</math> is [closed](/source/closed_and_exact_differential_forms) (for all ''<math>\xi</math>'' in <math>\mathfrak{g}</math>).

Suppose that <math>\iota_{\rho(\xi)} \omega \,</math> is not just closed but also exact, so that <math>\iota_{\rho(\xi)}\omega
=\mathrm{d}H_\xi</math> for some function <math>H_\xi : M \to \mathbb{R}</math>. If this holds, then one may choose the <math>H_\xi</math> to make the map <math>\xi \mapsto H_\xi</math> linear. A '''momentum map''' for the ''<math>G</math>''-action on <math>(M, \omega)</math> is a map <math>\mu : M \to \mathfrak{g}^*</math> such that

:<math>\mathrm{d}(\langle \mu, \xi \rangle) = \iota_{\rho(\xi)} \omega</math>

for all ''<math>\xi</math>'' in <math>\mathfrak{g}</math>. Here <math>\langle \mu, \xi \rangle</math> is the function from ''<math>M</math>'' to '''''<math>\mathbb{R}</math>''''' defined by <math>\langle \mu, \xi \rangle(x) = \langle \mu(x), \xi \rangle</math>. The momentum map is uniquely defined up to an additive constant of integration (on each connected component).

A <math>G</math>-action on a symplectic manifold <math>(M, \omega)</math> is called '''Hamiltonian''' if it is symplectic and admits a momentum map.

A momentum map is often also required to be '''<math>G</math>-equivariant''', where ''<math>G</math>'' acts on <math>\mathfrak{g}^*</math> via the [coadjoint action](/source/coadjoint_action), and sometimes this requirement is included in the definition of a Hamiltonian group action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the momentum map coadjoint equivariant. However, in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the [Euclidean group](/source/Euclidean_group)). The modification is by a 1-[cocycle](/source/Group_cohomology) on the group with values in <math>\mathfrak{g}^*</math>, as first described by Souriau (1970).

== Examples of momentum maps==
In the case of a Hamiltonian action of the circle <math>G = U(1)</math>, the Lie algebra dual <math>\mathfrak{g}^*</math> is naturally identified with <math>\mathbb{R}</math>, and the momentum map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when <math>M</math> is the [cotangent bundle](/source/cotangent_bundle) of <math>\mathbb{R}^3</math> and <math>G</math> is the [Euclidean group](/source/Euclidean_group) generated by rotations and translations. That is, <math>G</math> is a six-dimensional group, the [semidirect product](/source/semidirect_product) of <math>\operatorname{SO}(3)</math> and <math>\mathbb{R}^3</math>. The six components of the momentum map are then the three angular momenta and the three linear momenta.

Let <math>N</math> be a smooth manifold and let <math>T^*N</math> be its cotangent bundle, with projection map <math>\pi : T^*N \rightarrow N</math>. Let <math>\tau</math> denote the [tautological 1-form](/source/Tautological_one-form) on <math>T^*N</math>. Suppose <math>G</math> acts on <math>N</math>. The induced action of <math>G</math> on the symplectic manifold <math>(T^*N, \mathrm{d}\tau)</math>, given by <math>g \cdot \eta := (T_{\pi(\eta)}g^{-1})^* \eta</math> for <math>g \in G, \eta \in T^*N</math> is Hamiltonian with momentum map <math>-\iota_{\rho(\xi)} \tau</math> for all <math>\xi \in \mathfrak{g}</math>. Here <math>\iota_{\rho(\xi)}\tau</math> denotes the [contraction](/source/Interior_product) of the vector field <math>\rho(\xi)</math>, the infinitesimal action of <math>\xi</math>, with the [1-form](/source/1-form) <math>\tau</math>.

The facts mentioned below may be used to generate more examples of momentum maps.

===Some facts about momentum maps===
Let <math>G, H</math> be Lie groups with Lie algebras <math>\mathfrak{g}, \mathfrak{h}</math>, respectively.

# Let <math>\mathcal{O}(F), F \in \mathfrak{g}^*</math> be a [coadjoint orbit](/source/coadjoint_orbit). Then there exists a unique symplectic structure on <math>\mathcal{O}(F)</math> such that inclusion map <math>\mathcal{O}(F) \hookrightarrow \mathfrak{g}^*</math> is a momentum map.
# Let <math>G</math> act on a symplectic manifold <math>(M, \omega)</math> with <math>\Phi_G : M \rightarrow \mathfrak{g}^*</math> a momentum map for the action, and <math>\psi : H \rightarrow G</math> be a Lie [group homomorphism](/source/group_homomorphism), inducing an action of <math>H</math> on <math>M</math>. Then the action of <math>H</math> on <math>M</math> is also Hamiltonian, with momentum map given by <math>(\mathrm{d}\psi)_{e}^* \circ \Phi_G</math>, where <math>(\mathrm{d}\psi)_{e}^* : \mathfrak{g}^* \rightarrow \mathfrak{h}^*</math> is the dual map to <math>(\mathrm{d}\psi)_{e} : \mathfrak{h} \rightarrow \mathfrak{g}</math> (<math>e</math> denotes the [identity element](/source/identity_element) of <math>H</math>). A case of special interest is when <math>H</math> is a Lie subgroup of <math>G</math> and <math>\psi</math> is the [inclusion map](/source/inclusion_map).
# Let <math>(M_1, \omega_1)</math> be a Hamiltonian <math>G</math>-manifold and <math>(M_2, \omega_2)</math> a Hamiltonian <math>H</math>-manifold. Then the natural action of <math>G \times H</math> on <math>(M_1 \times M_2, \omega_1 \times \omega_2)</math> is Hamiltonian, with momentum map the direct sum of the two momentum maps <math>\Phi_G</math> and <math>\Phi_H</math>. Here <math>\omega_1 \times \omega_2 := \pi_1^*\omega_1 + \pi_2^*\omega_2</math>, where <math>\pi_i : M_1 \times M_2 \rightarrow M_i</math> denotes the projection map.
# Let <math>M</math> be a Hamiltonian <math>G</math>-manifold, and <math>N</math> a [submanifold](/source/submanifold) of <math>M</math> invariant under <math>G</math> such that the restriction of the symplectic form on <math>M</math> to <math>N</math> is non-degenerate. This imparts a symplectic structure to <math>N</math> in a natural way. Then the action of <math>G</math> on <math>N</math> is also Hamiltonian, with momentum map the composition of the inclusion map with <math>M</math>'s momentum map.

==Connection to Noether's Theorem==
[Noether's theorem](/source/Noether's_theorem) admits a particularly elegant formulation in terms of momentum maps. A brief summary of the relevant objects in this section: let symplectic manifold <math>(M, \omega)</math> be the [phase space](/source/phase_space) of a [Hamiltonian system](/source/Hamiltonian_system) with Hamiltonian <math>H:M\rightarrow\mathbb{R}</math>. Each point <math>z</math> in <math>M</math> represents a state of the system, and its time evolution is governed by <math>\dot z = X_H</math> where <math>X_H</math> is the [Hamiltonian vector field](/source/Hamiltonian_vector_field) corresponding to the Hamiltonian <math>H</math>; that is, <math>\iota_{X_H}\omega = dH</math>. Time evolution of functions <math>F:M\rightarrow\mathbb{R}</math> can be readily shown to be given by the [Poisson bracket](/source/Poisson_bracket) <math> \{F,H\} = \omega(X_F, X_H)</math>.

Now, Noether's theorem states that if the Hamiltonian is invariant under the (symplectomorphic) group action <math> \Phi(g,z): G\times M\rightarrow M</math> with infinitesimal generator <math>\rho(\xi)</math> as defined above, the corresponding momentum map <math>J(\xi)</math> will be a constant of motion. Proving this is simple: one simply differentiates the invariance condition <math>H(z) =H(\Phi(g,z))</math> with respect to <math>g</math> to get
<math display="block">\begin{align}
    0&=dH\cdot \rho(\xi)&\\
    \rightarrow\quad0&=\iota_{\rho(\xi)}\iota_{X_H}\omega\\
    \rightarrow\quad0&=\{H,J(\xi))\}\\
    \rightarrow\quad0&=\dot J(\xi)
\end{align}
</math>
===Example: Conservation of Angular Momentum===
Consider the classical [Kepler problem](/source/Kepler_problem). Here, the phase is the cotangent bundle of the plane. In Cartesian coordinates,
<math display="block"> H = \frac{1}{2}(p_1^2+p_2^2) -\frac{1}{\sqrt{q_1^2+q_2^2}}</math>
It is easy to see that the Hamiltonian is invariant under circular rotations of the plane. As mentioned earlier, the momentum map for the action on a cotangent bundle induced by an action on the base manifold is <math>\iota_{\rho(\xi)}\tau</math>. To compute this, we first note that <math>\tau</math> is given in coordinates by <math>p_1 dq_1+p_2dq_2</math>. Since there are no <math>dp_1</math> or <math> dp_2</math> terms in <math>\tau</math>, we actually only need to compute the part of <math>\rho(\xi)</math> lying in the base manifold <math>\mathbb{R}^2</math>, which is:
<math display="block">
\frac{d}{dg}\begin{bmatrix}\cos g & \sin g\\ -\sin g &\cos g\end{bmatrix}\begin{bmatrix}q_1\\q_2\end{bmatrix}\Big|_{g=0} =\begin{bmatrix}q_2\\-q_1\end{bmatrix} </math>
Contracting this with <math>\tau</math> yields <math>J = p_1q_2-p_2q_1</math>, and applying Noether's theorem tells us that this quantity, the angular momentum, is conserved throughout the course of the motion. This is equivalent to [Kepler's second law](/source/Kepler's_laws_of_planetary_motion).

== Symplectic quotients ==
Suppose that the action of a [Lie group](/source/Lie_group) ''<math>G</math>'' on the symplectic manifold <math>(M, \omega)</math> is Hamiltonian, as defined above, with equivariant momentum map <math>\mu : M\to \mathfrak{g}^*</math>. From the Hamiltonian condition, it follows that <math>\mu^{-1}(0)</math> is invariant under ''<math>G</math>''.

Assume now that ''<math>G</math>'' acts freely and properly on <math>\mu^{-1}(0)</math>. It follows that ''<math>0</math>'' is a regular value of <math>\mu</math>, so <math>\mu^{-1}(0)</math> and its [quotient](/source/Quotient_space_(topology)) <math>\mu^{-1}(0) / G</math> are both smooth manifolds. The quotient inherits a symplectic form from ''<math>M</math>''; that is, there is a unique symplectic form on the quotient whose [pullback](/source/pullback_(differential_geometry)) to <math>\mu^{-1}(0)</math> equals the restriction of ''<math>\omega</math>'' to <math>\mu^{-1}(0)</math>. Thus, the quotient is a symplectic manifold, called the '''Marsden–Weinstein quotient''', after {{harv|Marsden|Weinstein|1974}}, '''symplectic quotient''', or '''symplectic reduction''' of ''<math>M</math>'' by ''<math>G</math>'' and is denoted <math>M/\!\!/G</math>. Its dimension equals the dimension of ''<math>M</math>'' minus twice the dimension of ''<math>G</math>''.

More generally, if ''G'' does not act freely (but still properly), then {{harv|Sjamaar|Lerman|1991}} showed that <math>M/\!\!/G = \mu^{-1}(0)/G</math> is a stratified symplectic space, i.e. a [stratified space](/source/stratified_space) with compatible symplectic structures on the strata.

==Flat connections on a surface==
The space <math>\Omega^1(\Sigma, \mathfrak{g})</math> of connections on the trivial bundle <math> \Sigma \times G </math> on a surface carries an infinite dimensional symplectic form

:<math>\langle\alpha, \beta \rangle := \int_{\Sigma} \text{tr}(\alpha \wedge \beta).</math>

The gauge group <math> \mathcal{G} = \text{Map}(\Sigma, G) </math> acts on connections by conjugation <math> g \cdot A := g^{-1}(\mathrm{d}g) + g^{-1} A g </math>. Identify <math> \text{Lie}(\mathcal{G}) = \Omega^0(\Sigma, \mathfrak{g}) = \Omega^2(\Sigma, \mathfrak{g})^*</math> via the integration pairing. Then the map

:<math>\mu: \Omega^1(\Sigma, \mathfrak{g}) \rightarrow \Omega^2(\Sigma, \mathfrak{g}), \qquad A \; \mapsto \; F := \mathrm{d}A + \frac{1}{2}[A \wedge A]</math>

that sends a connection to its curvature is a moment map for the action of the gauge group on connections. In particular the [moduli space](/source/moduli_space) of flat connections modulo gauge equivalence <math>\mu^{-1}(0)/\mathcal{G} = \Omega^1(\Sigma, \mathfrak{g}) /\!\!/ \mathcal{G}</math> is given by symplectic reduction.

==See also==

* [GIT quotient](/source/GIT_quotient)
* [Quantization commutes with reduction](/source/Quantization_commutes_with_reduction)
* [Poisson–Lie group](/source/Poisson%E2%80%93Lie_group)
* [Toric manifold](/source/Toric_manifold)
* [Geometric Mechanics](/source/Geometric_Mechanics)
* [Kirwan map](/source/Kirwan_map)
* [Kostant's convexity theorem](/source/Kostant's_convexity_theorem)
* [BRST quantization](/source/BRST_quantization)

==Notes==
{{Reflist}}

==References==
* J.-M. Souriau, ''Structure des systèmes dynamiques'', Maîtrises de mathématiques, Dunod, Paris, 1970. {{issn|0750-2435}}.
* [S. K. Donaldson](/source/S._K._Donaldson) and [P. B. Kronheimer](/source/P._B._Kronheimer), ''The Geometry of Four-Manifolds'', Oxford Science Publications, 1990. {{isbn|0-19-850269-9}}.
* [Dusa McDuff](/source/Dusa_McDuff) and Dietmar Salamon, ''Introduction to Symplectic Topology'', Oxford Science Publications, 1998. {{isbn|0-19-850451-9}}.
*{{Citation |last1       = Choquet-Bruhat
 |first1      = Yvonne
 |authorlink = Yvonne Choquet-Bruhat
 |first2     = Cécile
 |last2      = DeWitt-Morette
 |title      = Analysis, Manifolds and Physics
 |publisher  = Elsevier
 |year       = 1977
 |location   = Amsterdam
 |isbn       = 978-0-7204-0494-4
 |url-access = registration
 |url        = https://archive.org/details/analysismanifold0000choq
}}
* {{cite book| last1=Ortega| first1=Juan-Pablo|last2=Ratiu| first2=Tudor S.| title=Momentum maps and Hamiltonian reduction|publisher = Birkhauser Boston|series=Progress in Mathematics|volume = 222|year = 2004|isbn = 0-8176-4307-9}}
*{{citation|last=Audin|first= Michèle|authorlink= Michèle Audin|title=Torus actions on symplectic manifolds|edition=Second revised|series= Progress in Mathematics|volume=93|publisher= Birkhäuser|year=2004|isbn=3-7643-2176-8}}
*{{citation|last1=Guillemin|first1=Victor|last2=Sternberg|first2=Shlomo|authorlink1=Victor Guillemin| authorlink2=Shlomo Sternberg|title=Symplectic techniques in physics|edition=Second |publisher=Cambridge University Press|year=  1990|isbn= 0-521-38990-9}}
*{{citation|first=Chris|last=Woodward|series=Les cours du CIRM|year=2010|volume=1|pages=55–98|publisher=EUDML|title=Moment maps and geometric invariant theory|arxiv=0912.1132|bibcode=2009arXiv0912.1132W}}
*{{citation|last=Bruguières|first=Alain|title=Propriétés de convexité de l'application moment|series=Séminaire Bourbaki|year=1987|journal=Astérisque |volume= 145–146|pages= 63–87|url=http://www.numdam.org/article/SB_1985-1986__28__63_0.pdf}}
*{{Citation |last1       = Marsden
 |first1      = Jerrold
 |authorlink1 = Jerrold E. Marsden
 |first2     = Alan
 |last2      = Weinstein
 |authorlink2= Alan Weinstein
 |title      = Reduction of symplectic manifolds with symmetry
 |journal    = Reports on Mathematical Physics
 |volume     = 5
 |year       = 1974
 |number     = 1
 |pages     = 121–130
 |doi = 10.1016/0034-4877(74)90021-4
 |bibcode = 1974RpMP....5..121M
 |url        = https://doi.org/10.1016/0034-4877(74)90021-4
|url-access= subscription
 }}
*{{Citation |last1       = Sjamaar
 |first1     = Reyer
 |first2     = Eugene
 |last2      = Lerman
 |title      = Stratified symplectic spaces and reduction
 |journal    = Annals of Mathematics
 |volume     = 134
 |year       = 1991
 |number     = 2
 |pages      = 375–422
 |doi = 10.2307/2944350
 |jstor = 2944350
 |url        = https://doi.org/10.2307/2944350
|url-access= subscription
 }}

{{DEFAULTSORT:Moment Map}}
Category:Symplectic geometry
Category:Hamiltonian mechanics
Category:Group actions

---
Adapted from the Wikipedia article [Momentum map](https://en.wikipedia.org/wiki/Momentum_map) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Momentum_map?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.