the second hundred · metaphor 165
Some sorrow doesn't drain away. It circles. The intensity holds steady while its shape keeps changing — hours of composure that curdle into a night of grief, then back to composure, and around again, at the same fixed pitch, for years.
We're taught to expect grief to be like a wound: raw, then scabbed, then scar, then nothing. A thing that runs downhill to rest. And often it is. But some grief refuses that arc. Nothing is healing and nothing is worsening — the total never moves — yet it is never still, because the same fixed store of feeling keeps trading places between held-together and undone.
Physics knows two kinds of motion, and only one of them comes to rest. A ball in mud stops; a planet does not. The difference is whether energy leaks out. Motion that conserves its energy — Hamiltonian motion — never settles. It orbits a fixed level of energy forever, position and momentum forever swapping.
The idea
Describe a swinging thing by two numbers: where it is (x, displacement from rest) and how fast it's going (p, momentum). Plot those two against each other and you get phase space — a map not of the thing but of its states. A single moment is a point; a life of motion is a curve threading through it.
A Hamiltonian system has a conserved quantity — an energy H(x, p) — that never changes as it moves. And that one fact fixes everything: the state can only travel along a curve of constant H, a level set. It cannot wander to a higher energy or drain to a lower one; it is trapped on its own ring. So it orbits. Forever. Not because something drives it, but because there is no exit from a closed contour.
The energy is stored in two forms that trade back and forth. At the bottom of the swing it's all motion (kinetic); at the top of the swing it's all height (potential), momentarily still. Neither form is created or destroyed — they pour into each other, and their sum is the invariant. Add damping and the spell breaks: energy leaks out each cycle, the ring shrinks into a spiral, and the motion finally comes to rest.
What to try
The instrument integrates a real pendulum with a symplectic scheme — the kind built to conserve energy honestly. With γ = 0, the state locks onto one faint ring and the energy trace below goes flat and stays flat; the energy drift chip shows the tiny residual — parts in a hundred-thousand — proving the flatness is computed, not faked. Press Stir to add momentum and it simply hops to a wider ring and orbits that one instead. There is no settling to be had.
Watch the exchange bar while it runs: the sorrow and composure segments slide against each other every cycle, one swelling exactly as the other shrinks, their total pinned. Push the energy up near E = 2 and you reach the separatrix — the state crawls agonizingly over the top, almost stalling. Now raise γ: the trace finally tips downward, the ring becomes an inward spiral, and the state drains to rest. Same system, one number changed — and grief that circled forever becomes grief that ends.
The mapping
The grief that worries us is not the loud kind that burns down to ash. It's the conserved kind. From the outside its level looks constant — the same held intensity, year over year — so people mistake it for stuck, or for indulged. But nothing is stuck. It is in constant motion; it's the total that won't move. What you read as steadiness is really an orbit: composure at the bottom of the swing, undoing at the top, endlessly pouring into each other.
This reframes what "getting over it" even means. Fading grief is dissipative — it needs a leak, some small loss of energy each cycle, and then time does the rest. Orbiting grief has no leak. You can't will the total down; you can only slowly, deliberately, install the damping — the ordinary frictions of a life that let a little feeling out each turn. Not a downhill roll but a patient bleed. And until it's installed, the swinging is not failure to heal. It's what conserved feeling does.
Read as life lessons
A constant intensity can hide ceaseless motion. The level set holds while the state races along it — what looks like being stuck is often an orbit.
Composure and grief are two coordinates of one quantity. When one rises the other falls by exactly as much; the trade is not progress, only phase.
Nothing conservative comes to rest. Rest requires dissipation — a small loss each cycle. You install that; you don't will the total down.
The mapping, exactly
| Mathematics | Life |
|---|---|
| the Hamiltonian H | The total charge of a grief — the invariant quantity of feeling that the motion carries. |
| the level set H = E | A fixed intensity of grief: the ring the state is condemned to travel while nothing dissipates. |
| potential energy | Sorrow — feeling held at height, pulled far from rest, momentarily motionless at the top of the swing. |
| kinetic energy | Composure in motion — feeling converted into carrying-on, fullest at the bottom of the swing. |
| the orbit | The circling: composure and sorrow trading, the same total returning to the same place, again and again. |
| damping γ | The ordinary frictions of living that finally let feeling out — the leak that turns orbit into spiral. |
| the fixed point at rest | Actually being done — the still center a dissipative grief spirals into, and a conservative one never reaches. |
The honest model
The state obeys a pendulum's Hamiltonian, H(x, p) = p²/2 + (1 − cos x), whose equations of motion are ẋ = p, ṗ = −sin x − γp. The first term of H is kinetic (composure), the second potential (sorrow); the page reads both off the live state and displays their exchange. Orbits are drawn as the actual constant-energy contours p = ±√(2(E − 1 + cos x)) for a spread of energies, with the separatrix at E = 2 — the boundary between swinging back and forth and rolling clean over the top — picked out in the tearing colour.
Conservation is not asserted; it's earned by the integrator. The state advances by a Störmer–Verlet (leapfrog) step, which is symplectic: with γ = 0 the energy neither grows nor drifts, and the readout tracks the true residual |E − E₀| so you can watch it stay microscopic. Turn γ up and the same integrator now bleeds momentum by e^(−γ·dt) each step; energy genuinely falls, the ring collapses to a spiral, and the state lands on the fixed point at the origin.
Where the metaphor tears
Perfect conservation is an idealization; the frictionless pendulum doesn't exist. Actual feeling nearly always has some leak, however slow, so most grief does fade — just far slower than we expect, which is a different diagnosis from "circles forever." The metaphor names a limiting case to make a point about steadiness; treat a real person as strictly Hamiltonian and you'll counsel patience where you should have looked for the leak.
An orbit that never decays can read as devotion, or as a trap. The mathematics is silent on which. A feeling held at fixed intensity for a lifetime might be love kept faith with, or a wound curated past all use — and nothing in the phase portrait tells them apart. The model describes the motion, never whether the motion is good.
The clean story has energy only leaking out. But lives are forced systems: an anniversary, a photograph, a smell drives fresh energy in and kicks the state to a wider ring. Real grief isn't a closed pendulum quietly winding down — it's a driven one, sometimes damped, sometimes stirred, and the neat conserved orbit is only the interval between shocks.