Definition of a moment map with physical context

Question

This was originally posted on Math Stack Exchange, but without an answer. I thus move it here, and hope it's not because I express it unclearly.

---

Suppose $(M,\omega)$ is a symplectic manifold "well" acted by a Lie group $G$. Then we can define the moment map $\mu: M \to \frak{g}^*$ to be the dual of $H_\xi$, which is defined to be the integral (if exists) of the (dual of the) vector field induced by the infinitesimal action of $G$. This makes sense to me mathematically.

According to this page, when $M$ is $T^*\mathbb{R}^3$ naturally acted by $SO(3)$, then the associated moment map reduces to the classical angular momentum (after choosing a suitable basis for $\frak{so}_3$). I also found an explicit calculation there. This makes sense to me physically.

What does not make sense to me is the passage from classical angular momentum to the modern definition of moment map. As pointed out in the last paragraph, the other side (modern->classical) can be easily obtained by choosing an ad-hoc basis. But from classical to modern is always harder. I wonder how they came up with the modern definition.

My guess

If I guess where it was from, I would start with Noether's theorem, which roughly says that any physical dynamic system with symmetries has a conserved quantity. Classical angular momentum is just one of the well-known conserved quantities. To unify all conserved quantities, people pinned down the axioms for the above to make sense, and defined the moment map to be a map that encodes a conversed quantity associate to a fixed symmetry.

Question

1. Is my guess correct?
2. If my guess is (closed to) correct, is there any text that develops the definition of moment maps in this vein?

Answer 1

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galileo group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galileo group, represented by the matrices: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the example of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the Galileo group. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized in diffeology [PIZ10], and then the classification theorem above has been extended in to any symplectic manifold [DIZ22]:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

---

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.