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

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.