The Top and the Trap

A heavy spinning top balanced upright and a charged particle held in the midplane of a Penning trap are both stabilized by a gyroscopic force against a potential that would otherwise push them away. Both lose that stability when two frequencies of opposite Krein signature collide: the top when its spin falls below a critical value, the trap when the magnetic field drops below its trapping limit. An exact linear dictionary maps the two problems onto each other, so the classical condition for the top to sleep becomes the condition for the trap to confine. Because both systems are axisymmetric and integrable, their nonlinear behaviour near the collision can be compared exactly. The sleeping Lagrange top always belongs to the soft type of the Hamiltonian–Hopf bifurcation; its normal-form invariants are fixed by geometry alone, and every such top has the same invariants at its critical spin. A Penning trap matches the top only when its octupole coefficient has the soft sign. Below the critical spin a top started near upright swings out to a cone whose half-angle is given exactly by a simple relation involving the spin ratio; the corresponding soft trap reaches a radius that agrees with this law to leading order. The next normal-form coefficient enters the scaled dynamics only as a relative correction of higher order. Including it reduces the discrepancy between the normal-form prediction and the top’s exact motion from one order to the next. Under isotropic damping the slow mode of both systems grows at the same rate, recovering the Kelvin–Tait–Chetaev fall of the top and the magnetron expansion of the trap as a single formula. When the control parameter is swept slowly through the collision, both systems obey the Airy equation in an appropriately scaled time, so the delay before a visible excursion appears scales in the same way. Rigid-body integrations of the top and direct integrations of the trap confirm each of these statements. What transfers is everything that depends only on the shared normal-form type and its invariants. What does not transfer is the global geometry beyond the normal-form neighbourhood, where the sphere caps the top’s tilt while the trap’s radial excursion continues to grow, and the hard bifurcation, which a trap can exhibit and a Lagrange top cannot.