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On Hammersley's minimum problem for a rolling sphere

Published online by Cambridge University Press:  24 October 2008

A. M. Arthurs  and
G. R. Walsh
A. M. Arthurs
Affiliation:
Department of Mathematics, University of York
G. R. Walsh
Affiliation:
Department of Mathematics, University of York
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Abstract

The problem posed by Hammersley (1983) of finding the shortest path along which a sphere can roll from one prescribed state to another is formulated by using quaternion calculus of variations and optimal control theory. This leads to a system of coupled nonlinear differential equations with prescribed end conditions. From the resulting expression for the curvature, it is shown that the differential equation of the required path in intrinsic coordinates is the same as the equation of motion of a simple pendulum, giving a solution in terms of elliptic integrals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 529 - 534
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Hammersley, J. M.. Oxford commemoration ball. In Probability, Statistics and Analysis. London Math. Soc. Lecture Note Series 79 (Cambridge University Press, 1983), 112–142.CrossRefGoogle Scholar
[2]Leitmann, G.. An Introduction to Optimal Control (McGraw-Hill, 1966).Google Scholar