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Noether's conservation law for Hamiltonian actions

Statement

Assume ${A C}_{ω}$ . Let $(M, ω, G, μ)$ be a Hamiltonian $G$ -space, and let $H \in C^{\infty} (M)$ be $G$ -invariant: $H (g \cdot p) = H (p)$ for all $g \in G$ and $p \in M$ . Then

${μ^{ξ}, H} = 0 for every ξ \in g,$

and the moment map $μ$ is constant along the Hamiltonian flow of $H$ . In the language of mechanics, every component of the moment map is a conserved quantity for the dynamics generated by the invariant Hamiltonian $H$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , a Hamiltonian $G$ -space $(M, ω, G, μ)$ and a $G$ -invariant smooth function $H$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through the fundamental-field interface cited in [F2].

[F1]

$H (g \cdot p) = H (p)$ for all $g \in G$ and $p \in M$ . [given]

[F2]

$ξ_{M} (p) = {\frac{d}{d t} ∣}_{0} \exp_{G} (- t ξ) \cdot p$ and $ξ_{M}$ is smooth. Fundamental vector fields for a left action.

[F3]

$X_{μ^{ξ}} = - ξ_{M}$ for every $ξ$ . Moment map components generate the negative infinitesimal action.

[F4]

${F, G} = ω (X_{F}, X_{G})$ and $X_{G} (F) = {F, G}$ . Poisson bracket on a symplectic manifold.

[F5]

$F$ is a first integral of $H$ , meaning constant along every integral curve of $X_{H}$ , if and only if ${F, H} = 0$ on $M$ . $F$ is a first integral of $H$ iff $F$ and $H$ Poisson commute.

Proof

technique · direct

1.1

Fix $ξ \in g$ and $p \in M$ . The curve $t \mapsto H (\exp_{G} (- t ξ) \cdot p)$ is constant by [F1], so its derivative at $t = 0$ vanishes. By [F2] that derivative is $d H_{p} (ξ_{M} (p))$ , hence $d H (ξ_{M}) = 0$ on $M$ .

F1F2given

2.1

Noether's identity follows: $0 = d H (ξ_{M}) = d H (- X_{μ^{ξ}}) = - X_{μ^{ξ}} (H) = - {H, μ^{ξ}}$ by [F3] and the Poisson convention [F4]; hence ${μ^{ξ}, H} = {H, μ^{ξ}} = 0$ by skew-symmetry.

step 1.1F3F4

3.1

By [F5] and step 2.1, every component $μ^{ξ}$ is a first integral of $H$ , that is constant along each integral curve of $X_{H}$ . Since the covector $μ (p)$ is determined by its finitely many components $μ^{ξ} (p)$ , the map $μ$ is constant along the Hamiltonian flow of $H$ .

step 2.1F5A1∎

Depends on

Used by

The reduced harmonic oscillator flow on projective space Example
Invariant Hamiltonians descend to reduced Hamiltonians Proposition

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)
Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)