120. Moffatt, H.K. (2000) Euler's disk and its finite-time singularity. Nature 404, 833-834.
Euler's diskis a toy disk that can be easily set in precessional spinning motion on a table. It is interesting from a mechanical point of view because it exhibits a finite-time singularity, viz. an infinite precessional angular velocity within a finite time, about 100 seconds. This infinity is in practise resolved in a matter of milliseconds. I sought to explain this behaviour through analysing the viscous dissipation in the thin layer of air trapped between the disc and the table. Other dissipative mechanisms are undoubtedly present, but only this one has been analysed in detail.
The 'finite-time singularity problem' in fluid mechanics relates to smoothness of solutions of the Navier-Stokes equations: can a singularity of vorticity appear at finite time starting from smooth initial conditions? We don't yet know the answer to this very fundamental question. In the meantime, it is useful to have the Euler disk as a table-top model of finite-time-singularity behaviour.
I bought this toy in 1999 as a Christmas present
for a grandchild; it turned out to be
too interesting to give away!
130. Moffatt, H.K. & Shimomura, Y. (2002) Spinning eggs: a paradox resolved. Nature 416, 385-386.
If you spin a hard-boiled egg fast enough on a smooth table, it will rise to the vertical. The explanation is remarkably subtle, depending as it does on consideration of the slipping friction between the egg and the table. In this short paper, Yutaka Shimomura and I gave a simple explanation on the basis of a 'gyroscopic approximation', valid for sufficiently fast spin. This approximation led to an understanding of the role of the 'Jellett constant', and to identification of the vertical spinning state as that which minimises energy for a prescribed value of this constant.
Subsequent developments may be found in:
Moffatt, H.K., Shimomura, Y. & Branicki, M. Proc.R.Soc.Lond. A, 460, 3643-3672.
Shimomura, Y., Branicki, M. & Moffatt, H.K. Proc. R. Soc. Lond. A, 461, 1753-1774; 462, 371-390.
150. Moffatt, H.K. & Tokieda, T. (2008) Celt reversals: a prototype of chiral dynamics. Proc. Roy. Soc. Edin. A, 138, 361-368.
The rattleback (or 'celt' to give it its technical name) is a canoe-shaped object with a curious property: it spins 'normally' in one direction, but in the opposite direction it wobbles vigorously and then reverses its spin direction. The explanation is to be found in the 'chiral' geometry of the rattleback: it is not the same as its mirror image. The phenomenon was first discussed by Walker (1896), but the full explanation has proved elusive - see for example Bondi (1986).
Tadashi Tokieda and I found a means of analysing this phenomenon, whose explanation again requires consideration of dissipative (frictional) effects associated with the spin. More than one reversal of spin direction is possible if the frictional effects are sufficiently weak.
G. T. Walker. 1896 On a dynamical top. Q. J. Pure Appl. Math. 28 , 175–184.
H. Bondi. 1986 The rigid body dynamics of unidirectional spin. Proc. R. Soc. Lond.
The mechanical toys described above were the subject of an Exhibit at the Royal Society Summer Science Exhibition 2007; Photo: with Tadashi Tokieda, Konrad Bajer and Michal Branicki at the Exhibition; Dynamics of SPIN posters designed by Andy Burbanks.
156. Goldstein, R.E., Moffatt, H.K., Pesci, A.I. & Ricca, R.L. (2010) Soap-film Möbius strip changes topology with a twist. Proc. Nat. Acad. Sc. USA, 107, 21979-21984.
If a circular wire is twisted and folded back on itself, then dipped in soap solution and withdrawn, a one-sided soap film in the form of a Möbius strip can be easily formed (Courant 1940). The question then is: what happens if the wire is now unfolded and untwisted back to its original circular form? At some stage in this process, the film jumps to a two-sided 'disc' spanning the wire. This jump takes place in about one hundredth of a second. The linking number of the Plateau border with the wire frame has to jump from 2 to zero when the surface becomes two-sided, a mechanism controlled by viscosity at the wire surface.
We have analysed this phenomenon both theoretically, and experimentally by high-speed photography. This is the start of a more comprehensive study of dynamical processes in soap films (minimum-area surfaces) spanning flexible wires that may be knotted and/or linked.
Courant, R. 1940 Soap film experiments with minimal surfaces. Am. Math. Mon. 47, 167–174.
Douglas, J. 1932 One-sided minimal surfaces with a given boundary. T. Am. Math. Soc. 34, 731–756..
For subsequent developments, see
Goldstein, R. E., McTavish, J., Moffatt, H. K. & Pesci, A. I. 2014 Boundary singularities produced by the motion of soap films. PNAS, 111 (23), 8339–8344.
170. Moffatt, P.G. & Moffatt, H.K. (2014) Giffen goods and their reflexion property. The Manchester School, 82, 129-142. doi: 10.1111/manc.12003
My son Peter is Professor of Econometrics at the University of East Anglia. Occasionally he consults me about mathematical problems that he runs into. This paper concerned Giffen goods which have the unusual property that the demand for a Giffen good increases as its price increases, other prices being held fixed, in contrast to 'normal' response. Consider, by way of example, a hypothetical Scot who spends all his disposable income on oatmeal and malt whisky, the latter being a relatively expensive luxury. Suppose that the price of oatmeal increases due to persistent rain and a poor harvest; our Scot can then afford less whisky, but, needing to maintain his calory-intake, must compensate by buying more oatmeal, which is in these circumstances a Giffen good.
Photo by Penelope
Our first problem was to devise a family of Utility Functions U corresponding to goods that have this Giffen property. The figure shows surfaces U(x,y,z) = cst. for a three-good situation, for which each good has the Giffen property in some region of the 'goods-space'.