The rotating saddle not only is an interesting system that is able to trap a ball near its saddle point, but can also intuitively illustrate the operating principles of quadrupole ion traps in modern physics. Unlike the conventional models based on the mass-point approximation, we study the stability of a ball in a rotating-saddle trap using rigid-body dynamics. The stabilization condition of the system is theoretically derived and subsequently verified by experiments. The results are compared with the previous mass-point model, giving large discrepancy as the curvature of the ball is comparable to that of the saddle. We also point out that the spin angular velocity of the ball is analogous to the cyclotron frequency of ions in an external magnetic field utilized in many prevailing ion-trapping schemes.

## I. INTRODUCTION

The rotating saddle has a number of intriguing mechanical properties such as the
counterintuitive stabilization of particles in the vicinity of its saddle point,^{1–4} the unexpected precession due to a
Coriolis-like force in the inertial reference frame,^{5} and so on. What endows this demonstration with an even more
profound meaning is the application of its underlying physical principle to the field of ion
trapping. As illustrated by Wolfgang Paul in his Nobel Lecture,^{6} by rotating or vibrating a “saddle-like” electrostatic
potential (a.k.a. the quadrupole potential), one can realize a stable equilibrium and thus
confine ions in a vacuum chamber. Such analogy between mechanics and electromagnetics
ingeniously interprets the trapping mechanism of the Paul trap—a prototype of the quadrupole
ion trap family—in an intuitive way, which touches the frontier of many areas among atomic
physics,^{7–11} plasma
physics,^{12–14} quantum
computation,^{15–17} and so on.

^{3,4}As a special case, the full set of conditions for stabilization was further applied to the saddle-shaped surface by Brouwer in his 1918 pioneering work.

^{4}The surface equation of a saddle one typically considers is

^{18}

^{,}

*a*(>0), and where

*x*,

*y*, and

*z*all possess the dimension of length. The saddle point of the surface located at

*x*=

*y*= 0 is an unstable equilibrium in the static case [see Fig. 1(a)]. But as first derived by Brouwer,

^{4}and subsequently discussed in many other papers,

^{2,3}a mass-point constrained on this saddle can be stabilized (prevented from slipping away from the equilibrium point) when the angular velocity Ω of the saddle exceeds a critical value Ω

_{crit}

*g*≈ 9.8 m/s

^{2}is gravitational acceleration.

Although this model is intuitive and appealing, it can be easily noticed that the
mass-point model exhibits some important deficiencies in accounting for many experimental
phenomena. First of all, it was demonstrated in early experiments^{1} that both the radius of the ball *R*, and the
curvature of the saddle 2/*a* can significantly influence the stability of
the system. But the stabilization condition Eq. (2), in which the radius *R* is apparently absent, cannot describe
the size effect of the balls. Second, most previous works treated the motion of the ball as
a two-dimensional problem, and therefore, the dynamic constraint for the rigid ball to stay
on the saddle surface was neglected. As a consequence, the high-speed instability—a
phenomenon frequently observed, where the ball jumps off the surface as the saddle rotates
fast enough—has not yet been fully explained.

In this paper, we establish a rigid-body model for the rotating-saddle problem starting
with a derivation of the rigorous equation of motion for a ball on a rotating surface of
arbitrary shape. We linearize the equations in the vicinity of the equilibrium point to
obtain the stabilization condition. The resulting lower limit of the rotating speed for the
onset of trapping is found to agree with that of the mass-point model as the radius *R* → 0. By investigating the interaction force between the saddle and the
ball, we explain why the ball tends to jump off the saddle when the rotating speed is high.
Finally, by comparing the rotating-saddle trap with several quadrupole ion-trapping schemes,
we present an appropriate electrical analog of our rigid body model.

The rotating saddle trap provides a fantastic teaching example in undergraduate classes. For a system that would be unstable in a static saddle-like potential, it illustrates how stabilization can be achieved by rotating the potential, either mechanically or electrically. The model may be demonstrated simply using the mass-point approximation, which can lead to a basic understanding of the trapping mechanisms. However, the rigid-body model provides a more accurate description of its dynamics, as well as a more challenging problem. The analogy between the mechanical trap and ion traps brings a connection between classroom demonstrations and the modern charged-particle capturing techniques, widely used in physics research today.

## II. MECHANICAL MODEL

We consider a rigid ball with radius *R* rolling on a saddle surface
rotating with angular velocity $\Omega $. The coordinate system is chosen to be fixed on the rotating
saddle as illustrated in Fig. 1(a); thus, we are able
to avoid many time-dependent terms but, as a consequence, we acquire centrifugal and
Coriolis forces in such a non-inertial frame.

*x*,

*y*, and

*z*directions, respectively. The total degrees of freedom of the system are reduced due to the constraints provided by the saddle surface. One of the constraints comes from the requirement that the ball stays on the surface of the saddle. The position vectors of the contact point

*O*and the ball's center of mass

*C*therefore have to satisfy

**r**

_{C},

**r**

_{O}, and

**n**denote, respectively, the center of mass position vector, the position vector of the contact point with the saddle surface, and the unit normal vector at the contact point (see Fig. 2). The explicit form of the normal vector in Cartesian coordinates can be calculated as

*F*(

*x*,

*y*,

*z*) is the surface equation described in Eq. (1).

*d*

**r**

_{C}/

*dt*and its spin $\omega $:

*O*, so that the torques provided by the supporting and the frictional forces vanish [see Fig. 1(b)]. But one should be very careful; the contact point

*O*is not a fixed point, and the commonly used torque equation

**M**

_{O}=

*d*

**L**

_{O}/

*dt*is not valid for our system. Instead, a modification term should be added as follows (see Appendix for derivation)

*O*, and the right-hand-side is composed of two terms including (i) the rate of change of the angular velocity about point

*O*, and (ii) the rate of change of the angular velocity due to the time-dependence (

*d*

**r**

_{O}/

*dt*≠ 0) of the reference point

*O*.

*O*in Eq. (7) is

*I*=

*αmR*

^{2}(

*α*= 2/5 for solid spheres) is the moment of inertia of the ball; the first term on the right-hand-side is the contribution from the orbital motion, and the second term arises from the spin of the ball about its mass center. Since the ball is constrained on the saddle surface, we also know that the velocity of the contact point

*O*is always parallel to the velocity of the center of mass

*C*, and therefore $ d r O / d t \xd7 d r C / d t = 0 $ in Eq. (7).

*O*. The torque provided by gravitational force is simply $ \u222b V ( r \u2212 r O ) \xd7 g \u2009 d m = m R n \xd7 g $, as in the inertial frame. Due to the position and velocity dependence of the inertia forces (centrifugal force and Coriolis force), these forces have inhomogeneous distributions on the ball. For an infinitesimal segment with mass

*dm*at position

**r**, the torques provided by centrifugal force and Coriolis force are

**x**from a mass segment (located at

**r**) to the spherical center (located at

**r**

_{C}):

**x**=

**r**–

**r**

_{O}–

*R*

**n**(see Fig. 2). It is not difficult to evaluate the body integral of Eq. (9) by integrating over

**x**, which yields

**M**

_{cen}is the same as that acting on a mass-point; however, an additional term $ I \omega \xd7 \Omega $ appears in the expression of

**M**

_{cor}. This extra term is due to the asymmetrical distribution of the Coriolis force over the ball, which distinguishes the rigid-body model from the conventional mass-point model. By substituting Eqs. (8) and (10) into Eq. (7), we finally get the equation of motion for the rigid ball moving on an arbitrary surface in the rotating frame

To sum up, we have collected nine independent variables for our system: the position of the
center of mass (*x _{C}*,

*y*,

_{C}*z*), the spin angular velocity of the ball (

_{C}*ω*,

_{x}*ω*,

_{y}*ω*), and the position of the contact point (

_{z}*x*,

_{O}*y*,

_{O}*z*) contained in the expression of the normal vector

_{O}**n**. Noticing that the geometric constraint Eq. (3) is a holonomic constraint with no velocity-dependence, we are able to eliminate the coordinates (

*x*,

_{C}*y*,

_{C}*z*), and therefore, the total degrees of freedom of the system are reduced to six.

_{C}For the pure-rolling case, the kinematic constraint Eq. (5) is a non-holonomic constraint with a velocity-dependent term *d***r**_{C}/*dt*.^{19,20} Since we cannot further eliminate
more independent variables with this non-integrable constraint, we have to simultaneously
solve the two differential vector equations [the constraint Eq. (5) and the governing Eq. (11)] to find the evolution of the six
coordinates (*x _{O}*,

*y*, and

_{O}*z*) and (

_{O}*ω*,

_{x}*ω*, and

_{y}*ω*). An example of the trajectory of the point

_{z}*O*of a pure-rolling ball is presented in Fig. 4(a).

## III. LINEARIZED GOVERNING EQUATIONS

The rigorous motion equations derived in Sec. II are
nonlinear due to the formation of the normal vector **n**. To analyze the stability
of the system, we linearize the system by expanding the variables in the vicinity of
equilibrium.

*ϵ*, we can neglect the

*z*component of

**r**

_{O}and linearize it to be $ r O \u2248 x O i + y O j $ (see Fig. 3). In addition, the expression of the unit vector

**n**can be linearized to be

**r**

_{C}, together with its

*n*th derivatives, can also be expressed in terms of

**r**

_{O}according to Eq. (3)

*O*(

*ϵ*) and using the identity $ k \xd7 ( v \xd7 k ) \u2261 v $ for an arbitrary vector

**v**perpendicular to $k$, we find, for the pure-rolling case

One can find that the spin angular velocity *ω _{z}* is constant in
both the pure-rolling and the frictionless-slipping cases under linear approximation. For
the pure-rolling case, from Eq. (5), we know
that

*ω*,

_{x}*ω*is small. In Eq. (16), they are further multiplied by

_{y}*r*, thus the effect of the horizontal angular velocity of the ball in this case can be neglected, whereas in the frictionless-slipping case,

_{cz}*ω*,

_{x}*ω*are constant. However, non-zero

_{y}*ω*,

_{x}*ω*will induce a constant force on the ball, so that it cannot be stabilized near the origin of the saddle. We therefore consider $ \omega \u2248 \omega z k $ for both cases.

_{y}*d*/

*dt*) in the laboratory-frame form to its rotating-frame form

^{21–23}

*e*with mass

*m*and velocity

**v**in the laboratory frame, it “feels” a Lorentz force $ e v \xd7 B $ in an external magnetic field $B$. Likely in our mechanical model, given a specific vertical spin-angular velocity $\omega $, the ball would “feel” a side force whose direction is perpendicular to its velocity in the laboratory frame as if a charged particle were “feeling” a Lorentz force in a magnetic field.

On a more physical level, this intriguing Lorentz-like driving term appears due to the fact that the spin-orbit interaction couples the spin of the pure-rolling ball to its external orbital motion around the equilibrium point. In Sec. VI, we will further discuss how this Lorentz-like term can mimic the magnetic field for a specific quadrupole ion trap.

*ABCD*coefficients for the pure-rolling and the frictionless slipping cases are shown in Table I. The correctness of these linearized equations is simply verified in Fig. 4(c) by numerically solving the trajectory of a pure-rolling ball and comparing the result with the rigorous one.

Coefficient . | Pure-rolling . | Frictionless-slipping . |
---|---|---|

A | $ ( a \u2212 2 R ) \Omega 2 + 2 \alpha R \omega z \Omega \u2212 2 g ( 1 + \alpha ) ( a \u2212 2 R ) $ | $ ( a \u2212 2 R ) \Omega 2 \u2212 2 g a \u2212 2 R $ |

B | $ ( 2 + \alpha ) ( a + 2 R ) \Omega \u2212 2 \alpha R \omega z ( 1 + \alpha ) ( a + 2 R ) $ | 2Ω |

C | $ ( a + 2 R ) \Omega 2 \u2212 2 \alpha R \omega z \Omega + 2 g ( 1 + \alpha ) ( a + 2 R ) $ | $ ( a + 2 R ) \Omega 2 + 2 g a + 2 R $ |

D | $ \u2212 ( 2 + \alpha ) ( a \u2212 2 R ) \Omega \u2212 2 \alpha R \omega z ( 1 + \alpha ) ( a \u2212 2 R ) $ | –2Ω |

Coefficient . | Pure-rolling . | Frictionless-slipping . |
---|---|---|

A | $ ( a \u2212 2 R ) \Omega 2 + 2 \alpha R \omega z \Omega \u2212 2 g ( 1 + \alpha ) ( a \u2212 2 R ) $ | $ ( a \u2212 2 R ) \Omega 2 \u2212 2 g a \u2212 2 R $ |

B | $ ( 2 + \alpha ) ( a + 2 R ) \Omega \u2212 2 \alpha R \omega z ( 1 + \alpha ) ( a + 2 R ) $ | 2Ω |

C | $ ( a + 2 R ) \Omega 2 \u2212 2 \alpha R \omega z \Omega + 2 g ( 1 + \alpha ) ( a + 2 R ) $ | $ ( a + 2 R ) \Omega 2 + 2 g a + 2 R $ |

D | $ \u2212 ( 2 + \alpha ) ( a \u2212 2 R ) \Omega \u2212 2 \alpha R \omega z ( 1 + \alpha ) ( a \u2212 2 R ) $ | –2Ω |

*R*→ 0), a regular precession is observed. The angle of precession within an interval of 33.5 s agrees very well with the value predicted by the mass-point model

^{5}[see Fig. 4(b)]

*R*/

*a*→ 0 [also see Eq. (19) and Table I for comparison]. This difference in orbital patterns can be explained by analyzing the eigenfrequencies of the system. For the frictionless-slipping case the without spin-orbit interaction, the magnitudes of the two pairs of the eigenfrequencies are close to each other when Ω

^{2}≫

*g*/

*a*, forming a precessing secular motion and a micro-oscillation. For the pure-rolling case with the strongest spin–orbit interaction, however, the eigenfrequencies are significantly different, making the trajectory more complex.

## IV. CRITICAL ANGULAR VELOCITY

*ω*and a particular rotating speed Ω, the four coefficients are fixed, and therefore yield four complex eigenfrequencies. We plot the loci of the four eigenfrequencies on the complex plane as parametric trajectories of the rotation speed Ω. As an example illustrated in Fig. 5, when the rotating speed of the saddle exceeds a certain value Ω

_{z}_{crit}(in this case greater than roughly 14 s

^{–1}), all of these complex frequencies have non-negative imaginary parts, and the system is stabilized in the vicinity of the equilibrium point.

_{crit}can even be derived analytically as follows. As mentioned earlier, the stability condition requires the imaginary parts of all of its four roots to be non-negative, leading to

*ω*≈ 0 due to rotational friction (

_{z}*ω*= Ω in the laboratory frame) when moving in the vicinity of the equilibrium point (see a illustrational video in supplementary material).

_{z}^{31}Also the radius of the ball

*R*cannot exceed

*a*/2; otherwise, it will be stuck on the surface. Using these two conditions associated with inequalities Eq. (24), the critical angular velocity Ω

_{crit}for the rotating-saddle to trap a rigid ball with radius

*R*turns out to be

^{–1}.

The fact that Ω_{crit} is identical for both the pure-rolling and the
frictionless-slipping cases is convenient for our experiments in that we cannot actually
control the ball to be rolling or slipping in real circumstances. In addition, since
Ω_{crit} is independent of coefficient *α*, the radial mass
distribution (i.e., whether the ball is solid or hollow) does not influence the stability of
the system.

To verify the critical angular velocity for confining rigid balls, we fabricated a
saddle-shaped surface with a 3D printer. The geometrical parameter *a* of the
saddle was set to be 0.159 m. The saddle was driven by a 24-V dc motor to rotate around the
vertical axis. By adjusting the voltage on the dc motor from 0 to 24 V, we were able to
control the rotating speed Ω of the saddle, which was further measured by a laser
tachometer.

We used polyfoam balls for our experiment. The advantage of using polyfoam balls is that their small weights would have little influence on the rotating speed Ω of the saddle. By carefully placing polyfoam balls with different sizes onto the center of the rotating saddle and recording their motions with a high-speed camera, we can measure the time the ball is trapped by the saddle. This procedure was repeated a number of times for the balls with radii of 3.30, 2.65, and 2.23 cm. The results are shown in Figs. 6(a)–6(c).

As can be seen in Figs. 6(a)–6(c), the trapping time
of the polyfoam balls dramatically increases after certain thresholds of rotating speed Ω.
These thresholds lies closer to the critical angular velocity Ω_{crit} predicted by
our rigid-body model (dashed line) than that predicted by the mass-point model (dot-dashed
line). Such a discrepancy between the rigid-body model and the mass-point model becomes
considerably larger as the radius of the ball increases [see Fig. 6(d)].

To better evaluate the rationality of our model, we plot $ \Omega crit 2 $ versus 1/(*a* – 2*R*), because
according to Eq. (25), $ \Omega crit 2 $ should be proportional to 1/(*a* –
2*R*) with the slope of 2*g*. The experimental critical
angular velocities are chosen to be those velocities for which the trapping time of the
polyfoam ball first exceeds 60*π*/Ω, roughly 10 s in our cases.

The final results are presented in Fig. 6(e). Compared with the mass-point prediction, much better agreements can be found between the experimental data and the theoretical prediction of rigid-body model.

## V. HIGH-SPEED INSTABILITY

By analyzing the eigenfrequencies of the linearized system, it has been noticed that a
mass-point constrained on non-symmetric rotating saddles (i.e., saddles possessing different
radii of curvature in two directions) would again lose stability as the rotating speed Ω
exceeds an upper bound.^{24} For symmetric
saddle, however, either the mass-point model or the rigid-body model till now is only able
to predict a lower bound for stabilizing the balls. But we do observe that the trapping time
of a polyfoam ball becomes considerably shorter as the rotating speed of our symmetric
saddle becomes high [see Figs. 6(a)–6(c)]—there seems
to be a “vague” upper limit of Ω to confine the ball in our saddle.

A typical example of such high-speed instability is recorded by a high-speed camera in Fig. 7 (see a high-speed video in supplementary
material).^{31} As can be seen, as the
rotating speed Ω reaches 43.6 s^{–1} (which is about three times greater than the
critical rotating speed 12.2 s^{–1}), the ball losses contact with the saddle
surface and jumps off.

Here, we come up with an important mechanism that can lead to such high-speed instability.
Glancing back over the dynamic model we have built, we find the interaction between the
saddle and the rigid ball does not allow the supporting force **F**_{n} to point in toward the surface. This means
that **F**_{n}⋅**n** < 0 is forbidden in real
circumstances: as the supporting force becomes less than zero, the ball simply loses contact
with and jumps off the saddle surface.

As an example, we track the supporting force provided by the saddle surface as a function
of time (see Fig. 8). We calibrate the rotating speed Ω
to be 20 and 30 s^{–1} which are all beyond Ω_{crit}. When Ω is 20
s^{–1}, the supporting force remains positive over time (see red line in Fig. 8). But when Ω reaches 30 s^{–1}, we find a
negative supporting force appears, which is actually forbidden in real cases (a surface
cannot “pull” an object to itself). The existence of this forbidden interaction is why
high-speed instability occurs.

The condition that yields high-speed instability is not only dependent on the geometric and kinematic parameters of the system but also on the initial conditions. For different rotating speed Ω of the saddle, we plot the feasible regions of the initial positions of the ball, within which the supporting force remains positive when evolving over time [see Fig. 9]. As Ω becomes larger, the feasible region becomes smaller, and it becomes less likely to put the ball within the feasible region in real experiments for it to be trapped. Other systematic influences such as air flow and some defect of the apparatus may also contribute to the instability of the ball in real experiments.

## VI. ELECTROMAGNETIC ANALOGY

The analogy with a ball rolling on a rotating saddle provides an intuitive way to interpret
the mechanism of confining ions with a type of quadrupole ion trap called the Paul trap.
However, it has been pointed out by several papers that this analogy, to some extent, is
quantitatively inaccurate on close scrutiny.^{2,25,26} In this section, we are going to review these quantitative
analyses, and compare them with the rigid-body model of rotating saddle trap to find a more
accurate mechanical correspondence.

^{6}Each rod electrode of the Paul trap is connected to its diagonally opposite one, and a sinusoidal voltage with amplitude

*V*

_{0}is applied between these two electrode pairs. With this configuration, a phase difference of

*π*is induced between the voltages on the electrode pairs, and an oscillating or “flapping” saddle-like quadrupole electric potential is generated in the vicinity of the central line on the

*xy*cross-section

^{8}whose expression is

*r*

_{0}is the distance between two diagonal electrodes and Ω is the circular frequency of the applied AC voltage. Although the Paul trap utilizes the same concept of stabilizing the equilibrium by varying the potential over time

^{5}as the rotating saddle, it was noticed

^{2,25}that the “flapping” quadrupole potential is not quantitatively the same as the rotating-saddle trap.

^{25,27–29}In this configuration, they utilized a three-phase AC source which is often used to generate a rotating electromagnetic field.

^{30}Each pair of rods is connected to an output wire of a three-phase AC source, and thus the quadrupole electric potential generated in the vicinity of the central line has the form

In addition to the rotating quadrupole field, the RRF trap also contains a constant magnetic field $B$ that further modulates the oscillating frequency of the plasma confined inside the trap. What is interesting is that the Lorentz force $ e v \xd7 B $ on every single ion in the RRF trap, as mentioned previously, is of the same form of the driving term $ \alpha \Lambda \u0303 ( v Lab \xd7 \omega ) $ that appeared in Eq. (16) for a pure-rolling rigid ball in a rotating saddle. This indicates that the spin angular velocity $\omega $ of a pure-rolling ball is an analog of the constant magnetic field $B$ in the RRF traps.

^{27}is the same as Eq. (19), with coefficients ABCD containing terms similar to those in Table I

*V*is the voltage of a static electric field for

_{t}*z*direction confinement,

*ω*=

_{B}*eB*/

*m*is the cyclotron frequency of ions in the magnetic field, and

*k*

_{1},

*k*

_{2}are two geometrical factors that are determined by the electrode configuration.

*α*) and defining $ p = \alpha \Lambda \u0303 11 , q = \alpha \Lambda \u0303 22 $, we obtain

*V*

_{0}acts just as the gravitational force

*g*, and the spin of the ball along

*z*direction

*ω*mimics the cyclotron frequency of charges in the magnetic field

_{z}*B*(

*ω*=

_{B}*eB*/

*m*). Therefore, we conclude that the motion of a pure-rolling ball in the rotating saddle can quantitatively mimic the behavior of the ions confined in the RRF trap with a constant magnetic field.

## VII. CONCLUSION

We built a rigid-body mechanical model for the rotating-saddle traps. We would like to end our article by reviewing the similarities and differences between our rigid-body model and the mass-point model.

A modified critical angular velocity for the saddle to stabilize rigid balls is derived,
taking the size of the ball into consideration [see Eq. (25)]. This critical angular velocity is identical for both the
pure-rolling and the frictionless-slipping balls, and tends to that predicted by the
mass-point model when *R* → 0.

We found that the mass-point model is a limit of the frictionless-slipping model when the
radius of the ball *R* → 0. They not only have the same orbital patterns [see
Fig. 4(b)], but also present exactly the same
precessional behavior. On the contrary, the orbital pattern of a pure-rolling ball, due to
the spin-orbit interaction, is different from the above two cases, even when the radius of
the ball is negligible.

Finally, we deduced that our rigid body model is a mechanical analog of the motion of ions
in an RRF trap with magnetic field. Under linear approximation, the motion of the ball in *z* direction can be neglected; thus, its constant spin angular velocity $ \omega = \omega z k $ fixed in the *z* direction is analogous to the
cyclotron frequency of charged particles in the magnetic field.

## ACKNOWLEDGMENTS

This work was partly funded by the National Natural Science Foundation of China (NNSFC-J1310026).

### APPENDIX: DERIVATION OF EQ. (7)

*N*small segments labeled by $ i = 1 , 2 , 3 , \u2026 , N $. The total torque on the rigid body with respect to the contact point

*O*is

**r**

_{Oi}is the position vector defined by (

**r**

_{i}–

**r**

_{O}),

**F**

_{i}is the external force on each segment, and

**F**

_{ij}is the internal force between segment

*i*and

*j*. According to Newton's third law, all internal forces cancel out upon summation. On further use of Newton's second law

**F**

_{i}=

*m*

_{i}d^{2}

**r**

_{i}/

*dt*

^{2}, we have

## REFERENCES

The general saddle–shaped surface equation should be *z* = *x*^{2}/*a* – *y*^{2}/*b* where *a* and *b* are positive constants. In this article, however, for the sake of
simplicity, we consider the special case *a* = *b* (symmetric saddle). [This is a loss of generality, so I have removed the statement that
it is not.]