Momentum Maps and Noether’s Theorem

Noether’s theorem is often stated as “continuous symmetries correspond to conserved quantities.” Symplectic geometry makes this statement precise: a momentum map connects the infinitesimal generators of a symmetry group to Hamiltonian functions.

1. Lie Group Actions

Let a Lie group $G$ act on a symplectic manifold $(M, \omega)$. An element $\xi \in \mathfrak{g}$ of its Lie algebra induces the fundamental vector field

$$\xi_M(x) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t = 0} \exp(t\xi) \cdot x .$$

If the group action preserves the symplectic form, then $\mathcal{L}_{\xi_M}\omega = 0$. Locally, $\iota_{\xi_M}\omega$ is therefore a closed one-form.

2. Momentum Maps

A momentum map is a map $J: M \to \mathfrak{g}^*$ such that, for every $\xi \in \mathfrak{g}$, the function

$$J_\xi(x) = \langle J(x), \xi \rangle$$

has $\xi_M$ as its Hamiltonian vector field. In other words,

$$\iota_{\xi_M}\omega = \mathrm{d}J_\xi .$$

3. Rotational Symmetry

The rotation group $\mathrm{SO}(3)$ in three dimensions acts on a particle’s phase space, and its momentum map is the angular momentum

$$J(q, p) = q \times p .$$

A central potential depends only on $|q|$, so it is rotationally invariant and angular momentum is conserved. The familiar vector formula is thus a concrete coordinate expression of a general geometric mechanism.