Mechanical analog of a Paul Trap particle confinement—a ball is trapped in a time-varying quadrupole gravitational potential.

**How it works:**

A large saddle shape (attached to a plywood disk) is mounted on a multi-purpose turntable. The saddle shape is essentially a quadrupole gravitational potential. Rotation of this potential subjects the ball to an alternating repulsive and attractive potential, much like the time-varying electric quadrupole potential of a Paul trap used in trapping single ions or electrons.

A 22 cm diameter rubber ball represents the particle we wish to confine. Rotational speeds up to 236 RPM are possible with the variable-speed motor drive. At low speeds the ball simply rolls off the saddle. Starting around 135 RPM and up to about 160 RPM, stability is achieved; as the ball begins to roll away from the center in any direction, it experiences a periodic force back toward the center as it encounters a "hill" twice every revolution of the turntable. As the ball rolls back and forth through the center of the saddle, it experiences a torque about its vertical axis and begins to spin about that axis, quickly approaching the saddle rotational speed (a little less due to slippage). The actual excursions about the center are typically one or two centimeters with occasional larger perturbations. Ultimately it spins like a top in the center of the saddle accompanied by small oscillations. The record so far is 15 minutes. The reason why, after being stable for so long, the ball very suddenly (catastrophically) loses it, is not clear.

The phenomenon of parametric excitation is one in which an oscillating system is influenced by periodically varying one of its parameters (the gravitational potential in this demonstration); this is *not* the same as an oscillator being driven by a periodic external force. In the case of parametric excitation, the equations of motion take on the form of the Mathieu differential equation. Unlike the driven harmonic oscillator, parametric resonance takes place at a frequency *twice* the natural frequency and within a certain range of frequencies about that value. The linear Mathieu equation does not have a closed form solution.

The following empirical specifications for the stability of a solution are in lieu of an exact functional dependence. The stability of the ball in this varying potential depends on (1) the rotational speed, (2) the relative curvatures (of the ball and the saddle shape), and (3) the ball's moment of inertia. The three variables are not independent. For a given saddle shape, the ball's radius must have the same order of magnitude as the radius of curvature of the saddle. ^{1} The moment of inertia of the ball is proportional to the square of its radius. The ball's moment of inertia and saddle curvature determine the period of oscillation about equilibrium; the 22 cm dia rubber ball has a frequency of oscillation of 1.22 ± 0.05 Hz. ^{2} The interval of rotational speeds leading to stability is centered about 147RPM (2.45 Hz) which is indeed twice the ball's natural frequency.

At higher rotational speeds the encounters with the hills impart a greater impulse to the ball, knocking it well beyond the center, only to have it encounter another hill at a higher level accompanied by a more violent interaction and very soon the ball is literally whacked out of the saddle. The nonlinear Mathieu equation reigns here, with chaotic motion being the solution.

**Setting it up:**

Being rather large, it is best to have the turntable on the floor. Demonstrate the lack of stability at low and high rotational speeds and then try your hand in the frequency range of stability. ^{3} Center the ball on the saddle and release it with a slight twist of the wrist so that it starts with a modest spin—this technique enhances the probability of success. Long-term stability seems to be quite sensitive to the initial conditions. *Practice before the lecture!* If the ball remains in the saddle for one to two minutes, the chances that it will remain ten to fifteen minutes are excellent.

**Comments:**

Like the Inverted Pendulum, this is a marvelous demonstration of parametric resonance and a wonderful visual mechanical analog of particle confinement, whether in Paul traps or cyclic particle accelerators. The reader is encouraged to look at Pinto and Winter/Ortjohann below for more examples and many useful references. We would like to acknowledge and thank Justin Georgi for solving the topological problem and making the saddle shape.

For more details on our apparatus see Rueckner, Georgi, Goodale, and Rosenberg, "Rotating saddle Paul trap," Am. J. Phys. **63**(2), 186-187 (1995).

**Related Stability Demonstrations:**

P.N. Murgatroyd, Am J Phys **62**, 281-282 (1994). The magnetic analogue of the inverted pendulum

C. Sacket, E. Cornell, C. Monroe, and C. Wieman, Am J Phys **61**, 304-309 (1993). A magnetic suspension system for atoms and bar magnets

F. Pinto, The Physics Teacher **31**, 336-346 (1993). Parametric Resonance: An Introductory Experiment

H. Winter and H.W. Ortjohann, Am J Phys **59**, 807 (1991). Simple demonstration of storing macroscopic particles in a "Paul trap"

L.W. Alvarez, R. Smits, and G. Senecal, Am J Phys **43**, 293 (1975). Mechanical analog of the synchrotron, illustrating phase stability and two-dimensional focusing

1 The reason for this is simple. If the ball is small compared to the radius of curvature of the saddle, it must roll quite far (relative to its own size) from the center before it experiences any appreciable restoring force back to the center. Generally speaking, displacements far from equilibrium do not effect simple harmonic motion -- an approximation one must secure here.

2 The only "natural" frequency in this system is that of the ball rolling back and forth in the saddle. This frequency was determined by fitting a sheet of posterboard into the saddle (so that the ball sits in a trough of the same curvature), and timing 10 oscillations of the ball in the trough.

3 The interval of rotational speeds (135-160 RPM) corresponds to turntable speed settings of 50-60; 55 = 147 RPM.