nLab
moment map

The momentum map

The momentum map

Idea

A momentum map is a dual incarnation of a Hamiltonian action of a Lie group (or Lie algebra) on a symplectic manifold.

An action of a Lie group GG on a symplectic manifold XX by (Hamiltonian) symplectomorphisms corresponds infinitesimally to a Lie algebra homomorphism from the Lie algebra 𝔤\mathfrak{g} to the Hamiltonian vector fields on XX. If this lifts to a coherent choice of Hamiltonians, hence to a Lie algebra homomorphism 𝔤(C (X),{,})\mathfrak{g} \to (C^\infty(X), \{-,-\}) to the Poisson bracket, then, by dualization, this is equivalently a Poisson manifold homomorphism of the form

μ:X𝔤 *. \mu : X \to \mathfrak{g}^* \,.

This μ\mu is called the momentum map (or moment map) of the Hamiltonian action. : Moment map is a misnomer and physically incorrect. It is an erroneous translation of the French notion application moment. See this mathoverflow question for the history of the name.

The name derives from the special and historically first case of angular momentum in the dynamics of rigid bodies, see Examples - Angular momentum below.

Definition

The Preliminaries below review some basics of Hamiltonian vector fields. The definition of the momentum map itself is below in Hamiltonian action and the momentum map.

Preliminaries

This section briefly reviews the notion of Hamiltonian vector fields on a symplectic manifold

The basic setup is the following: Let (M,ω)(M,\omega) be a symplectic manifold with a Hamiltonian action of a Lie group GG. In particular that means that there is an action ν:G×MM\nu\colon G \times M \to M via symplectomorphisms (diffeomorphisms ν g\nu_g such that ν g *(ω)=ω\nu_g^*(\omega) = \omega). A vector field XX is symplectic if the corresponding flow preserves (again by pullbacks) ω\omega. The symplectic vector fields form a Lie subalgebra χ(M,ω)\chi(M,\omega) of the Lie algebra of all smooth vector fields χ(M)\chi(M) on MM with respect to the Lie bracket.

By the Cartan homotopy formula and closedness dω=0d \omega = 0

Xω=dι Xω \mathcal{L}_X \omega = d \iota_X \omega

where X\mathcal{L}_X denotes the Lie derivative. Therefore a vector field XX is symplectic iff ι(X)ω=dH\iota(X)\omega = d H for some function HC (M)H\in C^\infty(M), usually called Hamiltonian (function) for XX. Here XX is determined by HH up to a locally constant function. Such X=X HX = X_H is called the Hamiltonian vector field corresponding to HH. The Poisson structure on MM is the bracket {,}\{,\} on functions may be given by

{f,g}:=[X f,X g] \{ f, g\} := [X_f,X_g]

where there is a Lie bracket of vector fields on the right hand side.

For (M,ω)(M,\omega) a connected symplectic manifold, there is an exact sequence of Lie algebras

0R(C (M),{,})χ(M,ω)0. 0 \to \mathbf{R}\to (C^\infty(M), \{-,-\}) \to \chi(M,\omega) \to 0 \,.

See at Hamiltonian vector field – Relation to Poisson bracket.

Hamiltonian action and momentum map

Let (X,ω)(X, \omega) be a symplectic manifold and let 𝔤\mathfrak{g} be a Lie algebra. Write (C (X),{,})(C^\infty(X), \{-,-\}) for the Poisson bracket Lie algebra underlying the corresponding Poisson algebra.

Definition

A Hamiltonian action of 𝔤\mathfrak{g} on (X,ω)(X, \omega) is a Lie algebra homomorphism

μ˜:𝔤(C (X),{,}). \tilde \mu \;\colon\; \mathfrak{g} \longrightarrow (C^\infty(X), \{-,-\}) \,.

The corresponding function

μ:X𝔤 * \mu \;\colon\; X \longrightarrow \mathfrak{g}^*

to the dual vector space of 𝔤\mathfrak{g}, defined by

μ:xμ˜()(x) \mu \;\colon\; x \mapsto \tilde \mu(-)(x)

is the corresponding momentum map.

Remark

If one writes the evaluation pairing as

,:𝔤 *𝔤 \langle -,-\rangle : \mathfrak{g}^* \otimes \mathfrak{g} \to \mathbb{R}

then the equation characterizing μ\mu in def. reads for all xXx \in X and v𝔤v \in \mathfrak{g}

μ(x),v=μ˜(v)(x). \langle \mu(x), v \rangle = \tilde \mu(v)(x) \,.

This is the way it is often written in the literature.

Notice that this in turn means that

μ˜(v)=μ *,v. \tilde \mu(v)= \mu^\ast \langle -,v\rangle \,.
Proposition

The following are equivalent

  1. the linear map underlying μ˜\tilde\mu in def. is Lie algebra homomorphism;

  2. its dual μ\mu is a Poisson manifold homomorphism with respect to the Lie-Poisson structure on 𝔤 *\mathfrak{g}^\ast.

Proof

This follows by just unwinding the definitions.

In one direction, suppose that μ˜\tilde \mu is a Lie homomorphism. Since the Lie-Poisson structure is fixed on linear functions on 𝔤 *\mathfrak{g}^\ast, it is sufficient to check that μ *\mu^\ast preserves the Poisson bracket on these. Consider hence two Lie algebra elements v 1,v 2𝔤v_1, v_2 \in \mathfrak{g} regarded as linear functions ,v i\langle -,v_i\rangle on 𝔤 *\mathfrak{g}^\ast. Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark

μ *{,v 1,,v 2} =μ *,[v 1,v 2] =μ˜([v 1,v 2]) ={μ˜(v 1),μ˜(v 2)} ={μ *,v 1,μ *,v 2} \begin{aligned} \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\} &= \mu^\ast \langle-,[v_1,v_2]\rangle \\ & = \tilde \mu([v_1,v_2]) \\ & = \{\tilde\mu(v_1), \tilde\mu(v_2)\} \\ & = \left\{ \mu^\ast \langle -,v_1\rangle, \mu^\ast \langle -,v_2\rangle \right\} \end{aligned}

and hence μ *\mu^\ast preserves the Poisson brackets.

Conversely, suppose that μ\mu is a Poisson homomorphism. Then

μ˜[v 1,v 2] =μ *,[v 1,v 2] =μ *{,v 1,,v 2} ={μ *,v 1,μ *,v 2} ={μ˜(v 1),μ˜(v 2)} \begin{aligned} \tilde\mu [v_1,v_2] &= \mu^\ast \langle -, [v_1,v_2]\rangle \\ & = \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\} \\ & = \left\{ \mu^\ast \langle -, v_1\rangle, \mu^\ast \langle -, v_2\rangle \right\} \\ & = \left\{ \tilde\mu(v_1), \tilde\mu(v_2) \right\} \end{aligned}

and so μ˜\tilde \mu is a Lie homomorphism.

Examples

Angular momentum

Consider the action of SO(3) on 3\mathbb{R}^3, which induces a Hamiltonian action on T * 3 3× 3T^*\mathbb{R}^3\cong\mathbb{R}^3\times\mathbb{R}^3 via

(q,p)ASO(3)(Aq,pA 1)(q,p)\xrightarrow{A\in\text{SO(3)}}(Aq,pA^{-1})

where qq is a column vector and pp is a row vector. Then the momentum map for this Hamiltonian action is

μ:T *( 3)𝔰𝔬(3) *,μ(q,p),θ(Ω 1 Ω 2 Ω 3)(q×p)θ\mu\colon T^*(\mathbb{R}^3)\to \mathfrak{so}(3)^*,\quad \left\langle\mu(q,p),\;\vec\theta\cdot\begin{pmatrix}\Omega_1\\\Omega_2\\\Omega_3\end{pmatrix} \right\rangle\to (\vec{q}\times \vec p)\cdot\vec{\theta}

where

Ω 1=(0 0 0 0 0 1 0 1 0),Ω 2=(0 0 1 0 0 0 1 0 0),Ω 3=(0 1 0 1 0 0 0 0 0)\Omega_1=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad\Omega_2=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad \Omega_3=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}

If we choose Ω 1,Ω 2,Ω 3\Omega_1,\Omega_2,\Omega_3 as an orthonormal basis of 𝔰𝔬(3)\mathfrak{so}(3) and then identify 𝔰𝔬(3)𝔰𝔬(3) * 3\mathfrak{so}(3)\cong\mathfrak{so}(3)^*\cong\mathbb{R}^3, then μ(q,p)=q×p\mu(q,p)=\vec q\times\vec p, which is the angular momentum.

Properties

Relation to conserved quantities

The values of the momentum map for each given Lie algebra generator may be regarded as the conserved currents given by a Hamiltonian Noether theorem.

Specifically if (X,ω)(X,\omega) is a symplectic manifold equipped with a “time evolution” Hamiltonian action 𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathbb{R} \to \mathfrak{Poisson}(X,\omega) given by a Hamiltonian HH and if 𝔤𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathfrak{g} \to \mathfrak{Poisson}(X,\omega) is some Hamiltonian action with momentum Φ(ξ)\Phi(\xi) for ξ𝔤\xi \in \mathfrak{g} which preserves the Hamiltonian in that the Poisson bracket vanishes

{Φ ξ,H}=0 \{\Phi^\xi, H\} = 0

then of course also the time evolution of the momentum vanishes

ddtΦ ξ={H,Φ ξ}=0. \frac{d}{d t} \Phi^\xi = \{H, \Phi^\xi\} = 0 \,.

See at Noether theorem – In terms of momentum maps/Hamiltonian Noether theorem.

Relation to constrained mechanics

In the context of constrained mechanics? the components of the momentum map (as the Lie algebra argument varies) are called first class constraints. See symplectic reduction for more.

The momentum map is a crucial ingredient in the construction of Marsden–Weinstein symplectic quotients and in other variants of symplectic reduction.

References

The concept is originally due to Jean-Marie Souriau.

General

Lecture notes and surveys include

Original articles include

Further developments are in

  • M. Spera, On a generalized uncertainty principle, coherent states and the moment map, J. of Geometry and Physics 12 (1993) 165-182, MR94m:58097, doi

  • Ctirad Klimcik, Pavol Severa, T-duality and the moment map, IHES/P/96/70, hep-th/9610198; Poisson-Lie T-duality: open strings and D-branes, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, hep-th/9512124

  • A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Math. Lec. Notes Series, AMS 1999, (pdf)

  • Friedrich Knop, Automorphisms of multiplicity free Hamiltonian manifolds, arxiv/1002.4256

  • W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257-293.

See also

Momentum maps in higher geometry, Higher geometric prequantum theory, are discussed in

Relation to symplectic reduction

Reviews include for instance

Relation to equivariant cohomology

Relation to equivariant cohomology:

Generalization: group-valued momentum maps

The relation between momentum maps and conserved currents/Noether's theorem is amplied for instance in

  • Huijun Fan, Lecture 8, Moment map and symplectic reduction (pdf)

In thermodynamics

Since momentum maps generalize energy-functionals, they provide a covariant formulation of thermodynamics:

  • Jean-Marie Souriau, Thermodynamique et géométrie, Lecture Notes in Math. 676 (1978), 369–397 (scan)

  • Patrick Iglesias-Zemmour, Jean-Marie Souriau Heat, cold and Geometry, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (web, pdf, doi:978-94-009-7022-9_5)

  • Jean-Marie Souriau, chapter IV “Statistical mechanics” of Structure of dynamical systems. A symplectic view of physics . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

  • Patrick Iglesias-Zemmour, Essai de «thermodynamique rationnelle» des milieux continus, Annales de l’I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 1-24 (numdam:AIHPA_1981__34_1_1_0)

  • Jean-Marie Souriau,Mécanique statistique, groupes de Lie et cosmologie. In Colloques int. du CNRS; numéro 237; Aix-en-Provence, France, 1974; pp. 24–28, 59–113. English translation by F. Barbaresco, April, 2020. Available online: https://www.academia.edu/42630654/Statistical_Mechanics_Lie_Group_and_Cosmology_

    1_st_part_Symplectic_Model_of_Statistical_Mechanics(access on 20 April 2020)

Review includes

  • Charles-Michel Marle, From tools in symplectic and Poisson geometry to Souriau’s theories of statistical mechanics and thermodynamics, Entropy 2016, 18(10), 370 (arXiv:1608.00103)

  • Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498. https://www.mdpi.com/1099-4300/22/5/498

  • Koszul, J.-L., Introduction to Symplectic Geometry, SPRINGER, 2019

Last revised on April 26, 2020 at 04:58:06. See the history of this page for a list of all contributions to it.