Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T20:50:30.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Trajectory singularities of general planar motions

Published online by Cambridge University Press:  14 November 2011

P. S. Donelan ,
C. G. Gibson  and
W. Hawes
P. S. Donelan
Affiliation:
School of Mathematical and Computing Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand
C. G. Gibson
Affiliation:
Department of Mathematical Sciences, University of Liverpool, PO Box 147, Liverpool L69 3BX, UK
W. Hawes
Affiliation:
School of Mathematics and Statistics, Middlesex University, Bounds Green Road, LondonN11 2NQ, UK
Get access Rights & Permissions [Opens in a new window]

Extract

Local models are given for the singularities that can appear on the trajectories ofgeneral motions of the plane with more than two degrees of freedom. Versal unfoldings of these model singularities give rise to computer-generated pictures describing the family of trajectories arising from small deformations of the tracing point, and determine the local structure of the bifurcation curves.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 37 - 55
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arnold, V. I.. Normal forms of functions in the neighbourhoods of degenerate critical points. Russ. Math. Surv. 29 (1974), 11–49.Google Scholar
2Arnold, V. I.. Wavefront evolution and equivariant Morse lemma. Commun. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
3Arnold, V. I.. Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Russ. Math. Surv. 34 (1979), 3–38.Google Scholar
4Bruce, J. W., Kirk, N. P. and Plessis, A. A. du. Complete transversals and the classification of singularities. Nonlinearity 9 (1996), 1–23.Google Scholar
5Bruce, J. W.. On the degree of determinacy of smooth functions. Bull. Lond. Math. Soc. 13 (1981), 55–58.CrossRefGoogle Scholar
6Bruce, J. W., Plessis, A. A. du and Wall, C. T. C.. Determinacy and unipotency. Invent. Math. 88 (1987), 521–554.CrossRefGoogle Scholar
7Cocke, M. W., Donelan, P. S. and Gibson, C. G.. Instantaneous singularity sets for planar and spatial motions. Preprint, University of Liverpool, UK (1998).Google Scholar
8Gaffney, T.. The structure of T A(f), classification and an application to differential geometry. In AMS Proc. of Symp. in Pure Mathematics. Part 1. Singularities 40 (1983), 409–427.CrossRefGoogle Scholar
9Gaffney, T., Plessis, A. A. du and Wilson, L. C.. Map-germs determined by their discriminant. In Proc. of the Hawaii–Provence Singularities Conf., Travaux en Cours (Paris: Hermann, 1990).Google Scholar
10Gaffney, T. and Wilson, L.. Equivalence theorems in global singularity theory. In AMS Proc. of Symp. in Pure Mathematics. Part 1. Singularities 40 (1983), 439–447.CrossRefGoogle Scholar
11Gibson, C. G.. Singular points of smooth mappings. In Research notes in mathematics (London: Pitman, 1979).Google Scholar
12Gibson, C. G. and Hobbs, C. A.. Local models for general one-parameter motions of the plane and space. Proc. R. Soc. Edinb. A 125 (1995), 639–656.CrossRefGoogle Scholar
13Gibson, C. G. and Hobbs, C. A.. Singularity and bifurcation for general two-dimensional planar motions. New Zealand J. Math. 25 (1996), 141–163.Google Scholar
14Gibson, C. G., Hobbs, C. A. and Marar, W. L.. On versal unfoldings of singularities for general two-dimensional spatial motions. Acta Applicandae Mathematicae 47 (1997), 221–242.CrossRefGoogle Scholar
15Gibson, C. G., Marsh, D. and Xiang, Y. Singular aspects of generic planar motions with two degrees of freedom. Preprint, Napier University (1996).Google Scholar
16Hawes, W.. Multi-dimensional motions of the plane and space. PhD thesis, University of Liverpool (1995).Google Scholar
17Hobbs, C. A. and Kirk, N. P.. On the classification and bifurcation of multigerms of maps from surfaces to 3-space. Preprint, Oxford–Brookes University (1996).Google Scholar
18Kirk, N. P.. Computational aspects of singularity theory. PhD thesis, University of Liverpool (1993).Google Scholar
19Martinet, J.. Singularities of smooth functions and maps. In London Mathematical Society Lecture Note Series, vol. 58 (Cambridge University Press, 1982).Google Scholar
20Mather, J. N.. Stability of C∞ mappings. VI. The nice dimensions. In Springer Lecture Notes in Mathematics, 192, Proc. Liverpool Singularities—Symposium 1 (1970), 207–253.Google Scholar
21Mond, D. M. Q.. On the classification of germs of maps from ℝ2 to ℝ3. Proc. Lond. Math. Soc. 50 (1985), 333–369.CrossRefGoogle Scholar
22Morin, B.. Formes canoniques des singularités d'une application différentiable. C. R. Acad. Sci. Paris 260 (1965), 5662–5665, 6503–6506.Google Scholar
23Plessis, A. A. du. On the determinacy of smooth map-germs. Invent. Math. 58 (1980), 107–160.CrossRefGoogle Scholar
24Plessis, A. A. du and Wilson, L. On right-equivalence. Math. Z. 190 (1985), 163–205.CrossRefGoogle Scholar
25Rieger, J. H. and Ruas, M. A. S.. Classification of A-simple germs from k n to k 2. Comp. Math. 79 (1987), 99–108.Google Scholar
26Wall, C. T. C.. Affine cubic functions. III. The real plane. Math. Proc. Camb. Phil. Soc. 87 (1980), 1–14.CrossRefGoogle Scholar
27Whitney, H.. On singularities of mappings of Euclidean space. I. Mappings of the plane to the plane. Ann. Math. 62 (1955), 374–410.CrossRefGoogle Scholar