A fold catastrophe potential illustrated with a pure mechanical apparatus

We present an apparatus consisting of a bar tightly connected to a disc, that acts as a pulley, free to rotate around its axis. A body is suspended by means of a thread that is wrapped around the collar of the pulley. Depending on the values of the masses and also on the geometric parameters of the various parts of the apparatus, the system may, or may not, stay in static equilibrium. When it moves, the system may perform stable oscillations around stability positions, or the body continuously falls down, while the pulley executes a sequence of complete rotations. We analyse these very diverse behaviours, basing the discussion on the shape of the potential energy, plotted as a function of the angular position of the system, stressing that the experiment provides a very good example of a fold catastrophe potential, besides being funny and pedagogically very valuable.


Introduction
We present a pure mechanical system whose dynamics is analysed and discussed by means of the so-called washboard potential.Interesting enough, the behaviour of this system can be quite different when small changes occur either in the initial kinematic conditions or in the potential parameters.The presentation, in the classroom, of this, so to say, catastrophic behaviour, may allow the students to realize that not all potentials are "well behaved".The work reported here is based on research done in collaboration with Rogério Nogueira and Bojan Golli.Here, and omitting the most technical parts which can be found in [1], the emphasis is on the pedagogical value of the system, which is a good example of a virtuous interaction between experiment, theory and modelling.
Potential energies of the type (arbitrary units) U (x) = x 3 + bx, where x is the dynamic variable and b is the "control parameter", are well known in catastrophe theory: U (x) is the potential of the so-called "fold catastrophe" [2].For the case b < 0, the potential has both a local minimum and a local maximum, but, when b > 0, the potential is a monotonic increasing function.These potentials, which show a very different behaviour depending on the value of one or more of their parameters, play an crucial role in physics and several examples can be found, for instance, in condensed matter physics (in the explanation of superconductivity ), in high energy physics (in the explanation of the mass of elementary particles through the so-called Higgs mechanism), etc.
As stated above, in the classroom this kind of potential can surprise and even amuse students [3], drawing the student's attention to the importance of an energetic approach in the analysis of many situations.In section 2, we describe the system more in detail, and establish the equations describing its motion.In section 3, we analyse the various motions in the context of the fold catastrophe potential.Finally, in section 4, we draw the conclusions and summarize the principal ideas.

Dynamics of the system
We constructed on purpose the experimental device used in this study, consisting of i) a pulley of mass µ and radius R (manufactured in a 3D printer) that freely rotates around its axis; ii) a (double) bar or ruler of length L and (total) mass M attached to the pulley; iii) and, finally, a body of mass m hanging at the extremity of a long enough thread that goes around the pulley several times.Figure 2 shows these three elements and some relevant forces responsible for the dynamics of each part of the system.On the periphery of the pulley acts the force − T , causing the rotation of the pulley.The other forces acting on the pulley are its weight and the reaction at centre, which are not represented in the figure.The sum of these three forces is zero (the pulley has no translational movement) but on the pulley there is a torque of magnitude T R relative to the axis of rotation.
The tension, T acts on the body hanging from the thread; the other force on this body is its weight, p.On the bar acts its weight, P = M g, and the force that keeps it fixed to the pulley (not shown in the figure).Altogether, one has to take into account: i) the translation motion of the body hanging from the thread; ii) and the rotation of the "pulley + ruler" around an axis perpendicular to the plane of the figure and passing through point O.The position of the system can be described by the sole variable x (the vertical position, as shown in the figure), or, equivalently, by the angle ϕ, also represented in the figure, since x = ϕR.The equation of motion for this angular variable is readily derived [1] and it is given by where I is the total moment of inertia of the system (receiving contributions from the disc, from the bar and from the body, respectively): I = 1 2 µR 2 + M L 2 3 + mR 2 ; the angle ϕ 0 in (1), such that sin ϕ 0 = 2mR M L , corresponds to the angular equilibrium position (no angular acceleration) of the system.At this angular position the hanging body is at position x 0 = Rϕ 0 .Of course, an equilibrium position only exists for This dimensionless quantity, A, is the control parameter mentioned in the next section.

Catastrophe -static and and dynamic study
In our analysis, the ideal case of no friction and no air resistance forces is assumed.In this case, we can write a potential energy for the system which is only of gravitational type.With respect to some arbitrary origin [U (0) = 0] for position x = 0 (hence ϕ = 0), it reads as where the first term refers to the suspended body and the second one to the bar.It is more convenient to work with the dimensionless quantities V (ϕ) = U (ϕ) mgR and A, the control parameter.In this case we can consider the potential as given by This potential, which is represented in Figure 3 for various A, is known in the literature as the tilted washboard potential [4], because of its shape, and it is pedagogically very rich, given the numerous analyses that can be carried out on it.We can find it in various situations even in basic physics [3,5,6].Let us first look at statics.Without the small body hanging from the thread, the system remains in equilibrium with the ruler in the vertical position, that is, for ϕ = 0.As we increase the weight of the body hanging from the string, the equilibrium angle, ϕ 0 , also increases.Static equilibrium still exists if the mass of the small body is such that m < M L/(2R) , that is, if A > 1.In this situation, e.g.A = 3 shown in Figure 3, the potential exhibits local maxima and local minima.In Figure 3, each minimum corresponds to an equilibrium position and we indicate one of such ϕ 0 .
A limiting situation occurs at A = 1, for which the maxima and minima disappear, merging each other and becoming inflection points.The equilibrium angular position approaches ϕ 0 = 90 o as A → 1 + .For A < 1, the static equilibrium is no longer possible.
In the real experiment, the most practical way to go from the equilibrium regime to the non-equilibrium regime is indeed by increasing the mass of the suspended body.In practice we do it by incorporating tiny grains of lead in the body.When ϕ 0 ∼ 90 o just by adding one more grain of lead, the systems de-stabilizes and the body falls down -we had reached A = 1 − , and the sudden change from stable to unstable is what we designate by catastrophe.
The catastrophic behaviour may also be recognized by studying the dynamics of the system, specially for A > 1.In Figure 4 we show the function V (ϕ), for A = 4.It is rather obvious that the potential near the minimum, ϕ 0 ∼ 14.5 o , exhibits a clear parabolic form.Defining the new angular variable φ = ϕ − ϕ 0 , the equation of motion (1), for the case of small oscillations around the minimum, φ ∼ 0 reduces to with the angular frequency squared explicitly given by ω 2 = gM L 2I cos ϕ 0 = gmR I √ A 2 − 1 .The comparison between the theoretical prediction for the period of the harmonic oscillations, T = 2π/ω, and the period actually measured for small oscillations is very nice [1].
However, if the energy is well above the local minimal energy, as E 1 in Figure 4, the system performs stable non-harmonic oscillations.The initial energy of the system can be easily varied by moving it away from the equilibrium position and/or by giving it an impulse, either on the hanging body or on the bar.For oscillations around ϕ 0 , there is a critical energy, E 2 , for the system to move in the stable mode.If the energy is increased, even infinitesimally, beyond this value, the oscillatory regime ceases and the body falls down.For even higher energies, such as E 3 , the only possible motion is of this very same type (unstable mode).It is amazing to observe this change from one mode to the other mode, just by changing the initial condition.And this is again another characteristic of fold catastrophe potentials.
A very nice computational simulation of the operation of our apparatus can be found in ref. [7].

Conclusions
We presented a mechanical device with a wonderful underlying physics.The associated potential is relevant because, on the one hand, it constitutes a clear pedagogical example of a potential exhibiting a fold catastrophe behaviour, which is very well captured by the students in the classroom or by the audience, in general, playing a curious motivational role.On the other hand, it turns out to be the potential that describes the so-called Josephson junctions (superconductor/thin insulator barrier/superconductor) in a circuit [8], which are the basis of a promising technology, among others, for the construction of qubits circuits for quantum computing machines [9].
This means that we start with the description of some funny apparatus with an amazing behaviour, and we may end up talking about quantum computers.Here, the main idea was to bring a simple concrete example of the cross-fertilization interaction between theory, experiment and computer modelling.

Figure 2 .
Figure 2. Representation of some forces acting on the system.

Figure 3 .
Figure 3.The washboard potential for three values of the control parameter, A. Only for A > 1 there are equilibrium positions.

Figure 4 .
Figure 4.The dimensionless potential for A = 4.Only for the total energy E 1 , stable oscillations are possible.For E > E 2 the body is bound to fall down.