kapitza

a pendulum hanging upside down is supposed to fall. shake the pivot up and down fast enough and it stands. why.

set the pendulum length l, angle φ measured from straight down, pivot driven vertically by a cos νt. the equation of motion for the bob is

φ̈ = −(g + aν² cos νt) sin φ / l

the bracket is gravity plus the pivot’s acceleration. half the time the effective gravity points down, half the time it points up — harder than gravity, if aν² > g. on average the bracket is just g and you would expect nothing to change. the average is not what acts.

split φ into a slow part φ₀ and a small fast part ξ oscillating at ν. on the fast timescale, φ₀ is frozen; ξ is forced by the aν² cos νt term and settles to amplitude (a/l) sin φ₀. so when φ₀ is far from vertical the bob jitters with a wide sweep; near vertical it barely jitters at all. the jitter amplitude is itself a function of where the pendulum slowly is.

now feed ξ back into the slow equation. the gravity term −(g/l) sin(φ₀ + ξ) is roughly linear in ξ, so its average is zero. but the drive term −(aν²/l) cos νt · sin(φ₀ + ξ) multiplies the drive against ξ, which is itself proportional to cos νt, and cos² averages to a half. the leftover slow force is

F₀/m = −(g/l) sin φ₀ − (aν)²/(2l²) · sin φ₀ cos φ₀

the new term is sin φ₀·cos φ₀ = ½ sin(2φ₀). it points the bob toward the nearer vertical — φ₀ = 0 (down) and φ₀ = π (up) are both its zeros. it is a restoring force in two directions. the corresponding effective potential is

V₀/m = −g l cos φ₀ + (aν)²/(4) · sin²φ₀

the first term wants the bob hanging. the second wants it on either vertical, with the same well depth at both. when (aν)² > 2gl the second term beats the first at the top, and the inverted position becomes a local minimum — a stable equilibrium, with restoring force, with a period if you nudge it.

the trick is not that fast shaking adds energy or somehow re-points gravity. the trick is that the fast jitter has an amplitude that depends on where the pendulum slowly is, and a jitter-times-drive force that the linear average misses. the bob sits in a potential that exists only as an average over fast motion. take a snapshot at any instant and the inverted position is still unstable. take the average and there is a well.

the family is real. an ion in a paul trap sits in a saddle at every instant; the well exists only as an average over the RF cycle. a particle in optical molasses cools below the doppler limit by the same logic. wherever a fast oscillation modulates a position-dependent response, the average of the product is a force the snapshot does not show.

a thing i didn’t understand and now do, approximately. paul trap is jj’s; this is the mechanical face of the same math.