Ergodic

#import "@preview/cetz:0.5.3": canvas, draw
#import draw: circle, content, line, rect
#set page(width: 660pt, height: auto, margin: 20pt, fill: none)
#set text(font: "Avenir Next", size: 14pt, fill: rgb("#19324f"))
#set par(leading: 0.55em)
#let (blue, green, ink, gray) = (rgb("#0b5fa5"), rgb("#286744"), rgb("#19324f"), rgb("#cdd3da"))
#let panel = block.with(
  width: 100%,
  inset: 12pt,
  radius: 7pt,
  fill: gray,
  breakable: false,
)

// A one-degree-of-freedom oscillator H = p^2/(2m) + k q^2/2 uses canonical coordinates (q,p),
// not two positions. Shell axes are the q and p intercepts of the H = energy ellipse.
#let shell_energy = 1
#let shell_width = .25
#let shell_axes(energy, mass: 1, stiffness: 1) = (
  calc.sqrt(2 * energy / stiffness),
  calc.sqrt(2 * mass * energy),
)
#let angle_point(time, ratio) = (calc.rem(time, 1), calc.rem(time * ratio, 1))
#let orbit(ratio, turns: 1) = canvas(length: 1pt, {
  let (width, height) = (210, 125)
  rect((0, 0), (width, height), fill: gray, stroke: rgb("#78828c") + .6pt)
  let previous = angle_point(0, ratio)
  for step in range(1, turns * 120 + 1) {
    let point = angle_point(step / 120, ratio)
    // Break at identified edges: never draw a spurious chord across a wrap.
    if calc.abs(point.at(0) - previous.at(0)) < .1 and calc.abs(point.at(1) - previous.at(1)) < .1 {
      let scaled = (previous, point).map(((x, y)) => (width * x, height * y))
      line(..scaled, stroke: blue + .65pt)
    }
    previous = point
  }
  content((width / 2, -15), $theta_1: 0 arrow.r 2pi$)
  content((-12, 62), rotate(-90deg, reflow: true)[$theta_2: 0 arrow.r 2pi$])
})
#panel[
  #text(size: 16pt, weight: "bold")[1  One oscillator: volume, contour, shell]
  #v(9pt)
  #grid(
    columns: (280pt, 1fr),
    gutter: 14pt,
    align: horizon,
    align(center, canvas(length: 1pt, {
      let (scale_q, scale_p) = (70, 45)
      // Outer edge of the green shell, then the blue H = E contour filling the interior
      for (energy, fill, stroke) in (
        (shell_energy + shell_width, rgb("#c6d8d2"), green + 1pt),
        (shell_energy, gray, blue + 1.5pt),
      ) {
        let (q_max, p_max) = shell_axes(energy)
        circle((0, 0), radius: (scale_q * q_max, scale_p * p_max), fill: fill, stroke: stroke)
      }
      for (tip, label, label-offset) in (((125, 0), $q$, (5, 0)), ((0, 83), $p$, (0, 23))) {
        line(tip.map(coord => -coord), tip, stroke: ink + .7pt, mark: (end: "stealth"))
        content((rel: label-offset, to: tip), label)
      }
      content((52, 20), $H < E$)
      // leader from just above the H=E label to its point on the contour
      let contour-label = (45, -84)
      content(contour-label, text(fill: blue)[$H=E$])
      line((rel: (0, 9), to: contour-label), (55, -53), stroke: blue + .7pt)
      content((-60, 90), text(fill: green)[$E < H < E+Delta E$])
    })),
    [
      $H(q,p) = p^2/(2m) + k q^2/2$
      #v(6pt)
      $q_(max)=sqrt(2E/k), quad p_(max)=sqrt(2 m E)$
      #v(6pt)
      Interior: $H < E$\ Blue contour: $H=E$\ Green shell: $E < H < E+Delta E$
      #v(6pt)
      Axes scaled independently; shell exaggerated.
    ],
  )
]
#v(12pt)
#grid(
  columns: (1fr, 1fr),
  gutter: 12pt,
  ..(
    ([2  Rational: closed orbit], 2, 1, [$omega_2 \/ omega_1=2$ · Opposite edges identified.]),
    (
      [3  Irrational: dense orbit],
      calc.sqrt(2),
      24,
      [$omega_2 \/ omega_1=sqrt(2)$ · Finite segment shown.],
    ),
  ).map(((title, ratio, turns, caption)) => panel[
    #text(size: 16pt, weight: "bold", title)
    #v(8pt)
    #align(center, orbit(ratio, turns: turns))
    #v(8pt)
    #align(center, caption)
  ]),
)
#v(12pt)
#block(
  width: 100%,
  inset: 10pt,
  radius: 5pt,
  fill: rgb("#c6d8d2"),
  breakable: false,
)[Irrational flow is ergodic on its *fixed-action torus* with uniform angle measure, not generally on the full energy surface.]