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The reduced harmonic oscillator flow on projective space

Example

Assume ${A C}_{ω}$ . Let $n \geq 1$ and $c > 0$ . On $C^{n}$ with $ω_{0} = \sum_{j} d x_{j} \land d y_{j}$ , the scalar circle action and its moment map $μ (z) = - \frac{1}{2} ∣ z ∣^{2} + c$ , consider the harmonic oscillator Hamiltonian $H (z) = \frac{1}{2} ∣ z ∣^{2}$ . It is circle invariant, so by Noether's theorem its flow preserves every level $μ^{- 1} (λ)$ , and on the zero level $S_{r}^{2 n - 1}$ it descends through the Hopf quotient ${C P}^{n - 1} = S_{r}^{2 n - 1} / S^{1}$ to the reduced Hamiltonian $h$ determined by $π^{*} h = ι^{*} H$ , where $ι : S_{r}^{2 n - 1} ↪ C^{n}$ and $π : S_{r}^{2 n - 1} \to {C P}^{n - 1}$ , with $r = \sqrt{2 c}$ . Because $H = c - μ$ on $C^{n}$ , the descended function is constant on the reduced space, and the projected flow of $X_{H}$ is trivial; the oscillator flow on the sphere moves along the circle orbits, which are exactly the fibres of the quotient. More generally, by the same proposition every circle-invariant Hamiltonian descends and its flow projects to the Hamiltonian flow of the descended function.

Facts & Assumptions

Given: ${A C}_{ω}$ , an integer $n \geq 1$ , a real number $c > 0$ , the scalar circle action on $C^{n}$ , its moment map $μ (z) = - \frac{1}{2} ∣ z ∣^{2} + c$ , the zero level $S_{r}^{2 n - 1}$ with $r^{2} = 2 c$ , and $H (z) = \frac{1}{2} ∣ z ∣^{2}$ .

[F1]

$μ$ is an equivariant moment map for the scalar circle action and the level $μ^{- 1} (0) = S_{r}^{2 n - 1}$ is a sphere with free circle action whose reduction is ${C P}^{n - 1}$ with the reduced form characterised by the pullback identity. Circle rotation on complex n-space and its quadratic moment map, Complex projective space as a circle symplectic reduction.

[F2]

If $H$ is $G$ -invariant then ${μ^{ξ}, H} = 0$ and $μ$ is constant along the flow of $H$ ; hence the flow preserves each level. Noether's conservation law for Hamiltonian actions.

[F3]

For an invariant Hamiltonian the restricted field $X_{H}$ is tangent to the level, projects to the Hamiltonian field of the descended function $h$ with $π^{*} h = ι^{*} H$ , and the restricted flow projects to the reduced flow. Invariant Hamiltonians descend to reduced Hamiltonians.

[A1]

The countable-choice assumption is The Axiom of Countable Choice ( ${A C}_{ω}$ ) and supplies the assumptions of [F2] and [F3].

[F4]

The Hamiltonian field is uniquely determined by $ι_{X_{H}} ω_{0} = d H$ (Hamiltonian vector fields exist uniquely for smooth functions).

Verification

technique · direct

1.1

The oscillator Hamiltonian is circle invariant, $H (e^{i θ} z) = H (z)$ , and satisfies $H = c - μ$ identically on $C^{n}$ because $μ = - \frac{1}{2} ∣ z ∣^{2} + c$ .

F1given

2.1

By [F2] the flow of $X_{H}$ preserves every level of $μ$ ; in particular it preserves the sphere $S_{r}^{2 n - 1}$ .

step 1.1F2

3.1

The zero level is regular and the circle action there is free by [F1]. It is proper because the action map has compact domain $S^{1} \times S_{r}^{2 n - 1}$ and Hausdorff target; inverse images of compact sets are closed in a compact space. On that sphere $H = c - μ$ restricts to $c$ , so [F3] gives the unique function $h$ with $π^{*} h = ι^{*} H$ , namely $h = c$ . Its differential is zero and nondegeneracy in [F4] gives $X_{h} = 0$ . For $n = 1$ the quotient is a point with this same constant function.

step 2.1F1F3F4

4.1

By [F3] the projected field $d π (X_{H})$ equals $X_{h} = 0$ , so the reduced flow is trivial. Directly, $d H = \sum_{j} (x_{j} d x_{j} + y_{j} d y_{j})$ and [F4] gives $X_{H} = \sum_{j} (y_{j} \partial_{x_{j}} - x_{j} \partial_{y_{j}})$ . Thus $\dot{z} = - i z$ and the flow is exactly $z (t) = e^{- i t} z (0)$ for all real $t$ , with no time rescaling. On the positive-radius sphere its trajectories are exactly the Hopf fibres. For an arbitrary smooth circle-invariant Hamiltonian, [F3] gives descent and projection on each integral curve interval; its reduced field need not vanish.

A1step 3.1F1F3F4∎

Depends on

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Sources

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