Abstract
At first glance it would appear that, when the dynamical equations of a system are known, we are able to accurately predict its state at any future moment in time. A closer examination, however, reveals the counter-example of molecular motion in gases. Although all the equations of motion of individual molecules and the laws of their collisions are known in this case, it is useless to solve these equations in an attempt to predict the precise positions and velocities of molecules at some future moment. The deterministic prediction fails in this case because of the extreme sensitivity of such a system to small variations in its initial conditions. The slightest perturbation in the coordinates and velocities of the molecules is sufficient to completely change their motion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Goldstein: Classical Mechanics (Addison-Wesley Press, Cambridge 1951)
E.T. Whittaker: Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge 1961)
P. Appell: Traité de Mécanique Rationnelle, Vols. 1 and 2 (Gauthier-Villars, Paris 1953)
V.I. Arnold: Mathematical Methods of Classical Mechanics (Springer, Berlin, Heidelberg 1989)
M. Henon, C. Heiles: Astron. J. 69, 73–79 (1964)
A.J. Lichtenberg, M.A. Lieberman: Regular and Stochastic Motion (Springer, Berlin, Heidelberg 1983)
P. Holmes: Physica D 5, 335–347 (1982)
R.Z. Sagdeev, D.A. Usikov, G.M. Zaslavsky: Nonlinear Physics (Harwood, New York 1988)
G.M. Zaslavsky, R.Z. Sagdeev: Introduction to Nonlinear Physics (Nauka, Moscow 1988, in Russian)
A.N. Kolmogorov: Dokl. Akad. Nauk SSSR 98, 527–530 (1954)
V.I. Arnold: Usp. Mat. Nauk 18, 13–40 (1963)
J.K. Moser: Nachr. Acad. Wiss. Gottingen, Math. Phys. K1 11a, 1–20 (1962)
V.I. Arnold, A. Avez: Ergodic Problems of Classical Mechanics (Benjamin, New York 1968)
J.K. Moser: Math. Ann. 169, 136–176 (1967)
V.I. Arnold: Dokl. Akad. Nauk SSSR 156, 9–12 (1964)
B.V. Chirikov: Phys. Rep. 52, 263–379 (1979)
R.S. MacKay, J.D. Meiss, I.C. Percival: Transport in Nonlinear Systems (Queen Mary College, London 1983)
P.J. Holmes, J.E. Marsden: J. Math. Phys. 23, 669–675 (1982)
A.N. Kolmogorov, S.V. Fomin: Elements of Function Theory and of Functional Analysis (Nauka, Moscow 1989)
R. Balescu: Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York 1975)
Ya.G. Sinai: Dokl. Akad. Nauk SSSR 153, 1261–1264 (1963)
Ya.G. Sinai: Usp. Mat. Mauk 25, 141–192 (1970)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mikhailov, A.S., Loskutov, A.Y. (1991). Unpredictable Dynamics. In: Foundations of Synergetics II. Springer Series in Synergetics, vol 52. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97294-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-97294-2_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-97296-6
Online ISBN: 978-3-642-97294-2
eBook Packages: Springer Book Archive