Abstract
All laws that describe the time evolution of a continuous system are given in the form of differential equations, ordinary (if the law involves one independent variable) or partial (if the law involves two or more independent variables). Historically the first law of this type was Newton's second law of motion. Since then Dynamics, as it is customary to name the branch of Mechanics that studies the motion of a body as the result of a force acting on it, has become the “typical„ case that comes into one's mind when a system of ordinary differential equation is given, although this system might as well describe any other system, e.g. physical, chemical, biological, financial etc. In particular the study of “conservative„ dynamical systems, i.e. systems of ordinary differential equations that originate from a time-independent Hamiltonian function, has become a thoroughly developed area, because of the fact that mechanical energy is very often conserved, although many other physical phenomena, beyond motion, can be described by Hamiltonian systems as well. In what follows we will restrict ourselves exactly to the study of Hamiltonian systems, as typical dynamical systems that find applications in many scientific disciplines.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
D.V. Anosov: Sov. Math. Dokl. 3, 1068 (1962)
D.V. Anosov: Sov. Math. Dokl. 4, 1153 (1963)
V.I. Arnold: Mathematical Methods of Classical Mechanics (Springer, Berlin, 1989)
V.I. Arnold and A. Avez: Ergodic problems of Classical Mechanics (Benjamin, New York, 1968)
Benettin G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn: Meccanica, March, 21 (1980)
Berry M.V. Berry: Regular and irregular motion. In: Topics in nonlinear dynamics, vol 46, ed by S. Jorna (AIP, New York, 1978) pp. 16-120
B.V. Chirikov: Sov. Phys. Dokl. 4, 390 (1959)
B.V. Chirikov: Phys. Rep. 52, 265, (1979)
G. Contopoulos: Bull. Astron. 3e ser. 2, Fasc. 1, 223 (1967)
G. Contopoulos: Order and Chaos in Dynamical Astronomy (Springer, Berlin, 2002)
J. Ford, S.D. Stoddard and J.S. Turner: Progr. Theoret. Phys. 50, 1547 (1973)
J. Ford: The Statistical Mechanics of Classical Analytical Dynamics. In: Fundamental Problems in Statistical Mechanics}, vol 3, ed by E.D.G. Cohen (North Holland, Netherlands, 1975) pp. 215-255
J. Ford: A Picture Book in Stochasticity. In: Topics in nonlinear dynamics}, vol 46, ed by S. Jorna (AIP, New York, 1978), pp. 121-146
Froeschle C. Froeschlé and J.P Scheidecker: Astrophys. Space Sci. 25, 373, 1973
Greene J. Greene: Ann. N.Y. Acad. Sci. 357, 80 (1980)
Gutzwiller Gutzwiller, M.C., Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990)
Henon M. Hénon and C. Heiles: AJ 69, 73 (1964)
J. Klafter, M.F. Shlesinger and G. Zumofen: Physics Today 49, 33 (1996)
J. Klafter, M.F. Shlesinger and G. Zumofen: Physics Today 55, 48 (2001)
A.N. Kolmogorov: Dokl. Akad. Nauk SSSR 124, 754 (1959)
Lichtenberg A.J. Lichtenberg M.A. and Lieberman: Regular and Chaotic Dynamics (Springer, Berlin, 1992)
Oseledec V.I. Oseledec: Trans. Moscow Math. Soc. 19, 197 (1968)
Ott E. Ott: Chaos in Dynamical Systems (Cambridge University Press) 1993
Ya.B. Pesin: Russ. Math. Surveys 32, 55 (1977)
Ya.G. Sinai: Sov. Math. Dokl. 4, 1818, (1963)
K. Tsiganis, H. Varvoglis and J. Hadjidemetriou: Icarus 146, 240 (2000)
H. Varvoglis, Ch. Vozikis and K. Wodnar: CMDA 89, 343, 2004
Zaslavsky G.M. Zaslavsky: Chaos in Dynamic Systems, 2nd edn (Harwood, London, 1987) pp 6–45
Author information
Authors and Affiliations
Editor information
Rights and permissions
About this chapter
Cite this chapter
Varvoglis, H. Regular and Chaotic Motion in Hamiltonian Systems. In: Dvorak, R., Freistetter, F., Kurths, J. (eds) Chaos and Stability in Planetary Systems. Lecture Notes in Physics, vol 683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10978337_2
Download citation
DOI: https://doi.org/10.1007/10978337_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28208-2
Online ISBN: 978-3-540-34556-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)