×

Topology and mechanics with computer graphics. Linear Hamiltonian systems in four dimensions. (English) Zbl 0617.58021

The aim of the paper is to show how the field of interactive computer graphics can contribute to new interactions between mathematics and mechanics, especially in the investigation of phenomena in the phase space (four or more dimensions) from the stable to the ergodic behavior. The authors start with a program on Liouville integrable quadratic Hamiltonians in two degrees of freedom (linear Hamiltonian systems in four dimensions) which can be solved exactly but have a relatively rich topological structure. I saw in Rio de Janeiro two really nice computer- animated color films produced by the authors on those linear systems. They also mention that other films on some nonlinear examples are currently in progress.
Reviewer: W.M.Oliva

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
68U99 Computing methodologies and applications
70-04 Software, source code, etc. for problems pertaining to mechanics of particles and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abraham, R.; Marsden, J., Foundations of Mechanics (1978), Benjamin-Cummings: Benjamin-Cummings Reading, Mass
[2] Arnold, V., Ordinary Differential Equations (1980), MIT Press: MIT Press Cambridge, Mass
[3] Arnold, V., Mathematical Methods of Classical Mechanics (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0386.70001
[4] Baxter, R.; Eiserike, H.; Stokes, A., A pictorial study of an invariant torus in phase space of four dimensions, (Weiss, Leonard, Ordinary Differential Equations, 1971 NRL-MRC Conference (1972), Academic Press: Academic Press New York), 331-349 · Zbl 0318.34056
[5] Churchill, R.; Kummer, M.; Rod, D., On averaging, reduction, and symmetry in Hamiltonian systems, J. Differential Equations, 49, 359-414 (1983) · Zbl 0476.70017
[6] Cushman, R., The momentum mapping of the harmonic oscillator, (Sympos. Math., 14 (1974)), 323-342
[7] Cushman, R., Topology of Integrable Hamiltonian Systems, (Lecture notes (1976), University of California: University of California Santa Cruz) · Zbl 0388.58008
[8] Cushman, R., Geometry of the energy momentum mapping of the spherical pendulum, CWI Newslett., 1, 4-18 (1983)
[9] R. Cushman, A. Kelley, and H. Koçak\(L_4\)J. Differential Equations; R. Cushman, A. Kelley, and H. Koçak\(L_4\)J. Differential Equations
[10] Cushman, R.; Rod, D., Reduction of the semisimple 1:1 resonance, Phys. D, 6, 105-112 (1982) · Zbl 1194.37125
[11] Duistermaat, J., Bifurcations of periodic solutions near equilibrium points of Hamiltonian systems, Centro Internazionale Matematico Estivo, Lecture notes (1983) · Zbl 0546.58036
[12] Dulac, V.; McIntosh, H., On the degeneracy of the two-dimensional harmonic oscillator, Amer. J. Phys., 33, 109-118 (1965)
[13] Feiner, S.; Salesin, D.; Banchoff, T., DIAL: A diagrammatic animation language, IEEE Comput. Graphics Appl., 2, No. 7, 43-56 (1982)
[14] Koçak, H., Linear Hamiltonian systems are integrable with quadratics, J. Math. Phys., 23, No. 12, 2375-2380 (1982) · Zbl 0507.70015
[15] Koçak, H., Normal forms and versal deformations of linear Hamiltonian systems, J. Differential Equations, 51, 359-407 (1984) · Zbl 0488.34034
[16] Moser, J., Integrable Hamiltonian systems, Progr. Math., 8, 233-289 (1980)
[17] Seifert, H.; Threlfall, W., (A Textbook of Topology (1980), Academic Press: Academic Press New York/London) · Zbl 0469.55001
[18] Smale, S., Topology and mechanics II, Invent. Math., 11, 45-64 (1970) · Zbl 0203.26102
[19] van der Meer, J. C., The Hamiltonian Hopf bifurcation, (Doctoral dissertation (1984), University of Utrecht) · Zbl 0853.58053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.