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

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

This $\mu$ 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 $(M,\omega)$ be a symplectic manifold with a Hamiltonian action of a Lie group $G$. In particular that means that there is an action $\nu\colon G \times M \to M$ via symplectomorphisms (diffeomorphisms $\nu_g$ such that $\nu_g^*(\omega) = \omega$). A vector field $X$ is symplectic if the corresponding flow preserves (again by pullbacks) $\omega$. The symplectic vector fields form a Lie subalgebra $\chi(M,\omega)$ of the Lie algebra of all smooth vector fields $\chi(M)$ on $M$ with respect to the Lie bracket.

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

$\mathcal{L}_X \omega = d \iota_X \omega$

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

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

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

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

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

Hamiltonian action and momentum map

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

Definition

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

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

The corresponding function

$\mu \;\colon\; X \longrightarrow \mathfrak{g}^*$

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

$\mu \;\colon\; x \mapsto \tilde \mu(-)(x)$

is the corresponding momentum map.

Remark

If one writes the evaluation pairing as

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

then the equation characterizing $\mu$ in def. reads for all $x \in X$ and $v \in \mathfrak{g}$

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

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

Notice that this in turn means that

$\tilde \mu(v)= \mu^\ast \langle -,v\rangle \,.$
Proposition

The following are equivalent

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

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

Proof

This follows by just unwinding the definitions.

In one direction, suppose that $\tilde \mu$ is a Lie homomorphism. Since the Lie-Poisson structure is fixed on linear functions on $\mathfrak{g}^\ast$, it is sufficient to check that $\mu^\ast$ preserves the Poisson bracket on these. Consider hence two Lie algebra elements $v_1, v_2 \in \mathfrak{g}$ regarded as linear functions $\langle -,v_i\rangle$ on $\mathfrak{g}^\ast$. Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark

\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 $\mu^\ast$ preserves the Poisson brackets.

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

\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 $\tilde \mu$ is a Lie homomorphism.

Examples

Angular momentum

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

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

where $q$ is a column vector and $p$ is a row vector. Then the momentum map for this Hamiltonian action is

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

where

$\Omega_1=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad\Omega_2=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad \Omega_3=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}$

If we choose $\Omega_1,\Omega_2,\Omega_3$ as an orthonormal basis of $\mathfrak{so}(3)$ and then identify $\mathfrak{so}(3)\cong\mathfrak{so}(3)^*\cong\mathbb{R}^3$, then $\mu(q,p)=\vec q\times\vec p$, 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 $(X,\omega)$ is a symplectic manifold equipped with a “time evolution” Hamiltonian action $\mathbb{R} \to \mathfrak{Poisson}(X,\omega)$ given by a Hamiltonian $H$ and if $\mathfrak{g} \to \mathfrak{Poisson}(X,\omega)$ is some Hamiltonian action with momentum $\Phi(\xi)$ for $\xi \in \mathfrak{g}$ which preserves the Hamiltonian in that the Poisson bracket vanishes

$\{\Phi^\xi, H\} = 0$

then of course also the time evolution of the momentum vanishes

$\frac{d}{d t} \Phi^\xi = \{H, \Phi^\xi\} = 0 \,.$

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.

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)

• Charles-Michel Marle, On Gibbs states of mechanical systems with symmetries, arXiv:2012.00582v2 [math.DG], Januray 13th 2021

• 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

• Barbaresco F. (2021) Archetypal Model of Entropy by Poisson Cohomology as Invariant Casimir Function in Coadjoint Representation and Geometric Fourier Heat Equation. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol 12829. Springer

• Barbaresco, F. Koszul lecture related to geometric and analytic mechanics, Souriau’s Lie group thermodynamics and information geometry. Info. Geo. 4, 245–262 (2021)

• Barbaresco F. (2021) Jean-Marie Souriau’s Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum. In: Barbaresco F., Nielsen F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer

• Marle CM. (2021) Gibbs States on Symplectic Manifolds with Symmetries. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol 12829. Springer

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