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The Persistence of Spin

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The Persistence of Spin The persistence of spin H.K.Moffatt Trinity College, Cambridge, CB2 1TQ, UK Abstract: Three spinning toys that illustrate fundamental dynamical phenomena are discussed: the rising egg, prototype of dissipative instability; the shuddering Euler's disc, prototype of finite- time singularity; and the reversing rattleback, prototype of chiral dynamics. The principles un- derlying each phenomenon are discussed, and attention is drawn to analogous phenomena in the vorticity dynamics of nearly inviscid fluids. 1 Introduction This informal evening lecture, delivered at the Carlsberg Academy, was con- cerned with dynamical principles illustrated by the behaviour of three mechan- ical toys. The first of these, the tippe-top, was the subject of a famous 1954 photograph (figure 1) of Niels Bohr (then honorary resident of the Carlsberg Academy) and Wolfgang Pauli, who were evidently intrigued by the tippe- top's ability to turn upside-down as it spins, a vivid example of a dissipative (or slow) instability driven by slipping friction at the point of contact with the floor. The rising egg (the spinning hard-boiled egg that rises to the vertical if spun rapidly enough) is a similar phenomenon, which may be understood in terms of minimising the spin energy subject to conservation, not of angular momentum, but rather of the `adiabatic Jellett invariant' (see section 2). My second toy is `Euler's disc' (see <www.eulersdisc.com>), that rolls on its bevelled edge on a plane table, and settles to rest (like a spun coin) with a rapid shudder, a beautiful table-top example of a finite time singularity. Controversy surrounds the question of what is the dominant contribution to the rate of dissipation of energy for this toy; but whatever mechanism is invoked, the singularity is resolved in the final split second of the motion, most probably by loss of contact between the disc and the table and consequent release of the no-slip constraint. My third toy is the rattleback, a canoe-shaped object that spins reasonably smoothly in one direction, but which, when spun in the opposite direction, exhibits a pitching instability leading to spin reversal. The rattleback is in 2 H.K.Moffatt Fig. 1. Niels Bohr and Wolfgang Pauli investigate the behaviour of the tippe-top; Photograph (1954) by Erik Gustafson, courtesy AIP Emilio Segre Visual Archives, Margrethe Bohr Collection fact very slightly deformed giving it a chiral (i.e. non-mirror-symmetric) mass distribution. It is this chiral property in conjunction with spin that leads to the intriguing behaviour, which may be regarded as providing a prototype of `chiral dynamics'. The nature of the dissipative effects that damp the motion are as yet ill-understood; but a semi-empirical choice of linear damping coeffi- cients provides a model that agrees well, at least qualitatively, with observed behaviour. The fluid counterpart of spin is of course vorticity, and it is therefore perhaps appropriate that, at this Symposium commemorating Helmholtz's seminal 1858 paper on the laws of vortex motion, the phenomenon of spin may be taken as a natural starting point. Just as friction plays a key role in each of the above toy examples, so frictional (i.e. viscous) effects in fluids are nearly always important no matter how small the viscosity may be, a fact that should be constantly borne in mind (as Helmholtz was himself well aware) when adopting inviscid modelling techniques. 2 The rising egg Spin a hard boiled egg sufficiently rapidly on a table and it will rise to the vertical, a parlour trick that invariably provokes a startled response: what is The Persistence of Spin 3 it that makes the centre-of-mass rise in this way? The key to understanding this phenomenon was provided by Moffatt & Shimomura (2002) (and in the comprehensive treatment of this problem by Moffatt, Shimomura & Branicki 2004, Shimomura et al. 2005, Branicki et al. 2006, and Branicki & Shimo- mura 2007; see also Bou-Rabee et al. 2005) on the basis of the `gyroscopic approximation', which is applicable when the spin is large and the slipping friction at the point of contact with the table weak. These conditions are the counterparts of the conditions of small Rossby number and large Reynolds number in the mechanics of rotating fluids. They imply a state of `balance' in which (at leading order) coriolis forces dominate the behaviour. Nevertheless, the frictional force at the point of slipping contact with the table exerts a torque relative to the centre-of-mass of the egg, causing this centre-of-mass to rise on a slow (dissipative) time-scale. Figures 2,3 and 4 show three posters prepared for the Summer Science Exhibition held at the Royal Society, London, in July 2007. These posters were designed by Andrew Burbanks (University of Portsmouth). Their purpose was to make the scientific principles that govern the behaviour of such spinning toys accessible to a wide public. The first poster (Figure 2) was based on photographs provided by Yutaka Shimomura of the rising egg phenomenon, including side reference to the analogous role of coriolis forces in large scale atmospheric dynamics. Under the gyroscopic approximation, the equations governing the spin of the egg admit an invariant J = Ωh, where Ω is the component of angular velocity about the vertical, and h is the height of the centre-of-mass above the table. This invariant similarly exists for the case of any convex axisymmetric body spinning with slipping friction on a horizontal plane boundary. For the particular case of a body that is part spherical, this invariant was found by Jellett(1872) (and so is appropriately called the Jellett invariant), and in this case it is exact (without need to make the gyroscopic approximation.) This is the case for the tippe-top (a mushroom-shaped spherical-cap top) which so intrigued Niels Bohr and Wolfgang Pauli; the manner in which the tippe-top turns upside-down when it spins was successfully explained by Hugenholtz (1952). The existence of the Jellett invariant provides a reasonably transparent explanation for the rise of the egg. The slipping friction causes a slow loss of energy (to heat), and the system seeks a minimum energy state compatible with the `prescribed' and constant value of J. At high spin rates (as required for the gyroscopic approximation), the energy is predominantly the kinetic energy of spin (the potential energy being much smaller). Consider the case of a uniform prolate spheroid with semi-axes a; b with a > b, the kinetic energy when the axis is horizontal is 2 2 2 2 E1 = (5=2)J =(a + b )b ; (1) and the kinetic energy when the axis is vertical is 4 H.K.Moffatt 2 2 2 E2 = (5=4)J =a b : (2) Hence, since a > b, it follows that E2 < E1, i.e the vertical state has lower en- ergy (for the same value of J) and is therefore the preferred stable equilibrium. Note that for the case of an oblate spheroid (a < b), the opposite conclusion holds: if spun sufficiently rapidly on the table about its axis of symmetry, it will rise to spin on its rim. A `go'-stone, or indeed a mint imperial, provides a suitably oblate object on which to experiment. The above argument is reminiscent of that used to explain why a body rotating freely in space will tend, under the effects of weak internal friction, to rotate about its axis of greatest inertia. For a prolate spheroid, this is an axis perpendicular to the axis of symmetry, a conclusion quite the opposite of that obtained above for the spinning egg problem. So why the difference? It is because in the case of the freely rotating body, angular momentum is conserved, and energy is minimised subject to this constraint. For the spinning egg, angular momentum is not conserved, but the Jellett invariant takes its place; this makes all the difference. The moral is: for any weakly dissipative system, first determine what is conserved (even in some approximate sense), then minimise energy subject to this constraint. The rise of the egg is only one ‘filtered’ aspect of the fifth-order nonlin- ear dynamical system that governs the non-holonomic rigid body dynamics. Superposed on the `slow' rise are rapid oscillations whose amplitude is con- trolled by the current state of spin. For sufficiently large spin (well above that required to make the egg rise) these oscillations can cause the normal reaction between egg and table to fluctuate to such an extent that the egg momentar- ily loses contact with the table, a prediction verified experimentally by Mitsui et al(2006). The analogous oscillations in the geophysical context are Rossby waves superposed on the mean large-scale atmospheric circulation. Dissipative instabilities have been set in an abstract geometric context by Bloch et al. (1994). The rising egg provides a prototype dissipative instability, brought to life by this simple table-top demonstration. 3 Euler's Disc The toy known as Euler's disc is a heavy steel disc that can be rolled on its edge on a horizontal surface (or on the slightly concave dish that is supplied with it, see figure 3). It works well on a glass-topped table. This toy exhibits a ‘finite-time singularity' in the final stage of its motion before it comes to rest on the table: the point of rolling contact of the disc on the table describes a circle with angular velocity Ω that increases `to infinity' as the angle α between the plane of the disc and the table decreases to zero, as it obviously must in this final stage of motion. I place `to infinity' in parentheses because in reality this potential singularity must be `resolved' in some way by physical processes during the final split second when the motion is arrested.
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