Abstract
A classical approach to the dynamics of Hamiltonian systems (or dynamical systems in general) is based on the notion of a phase space (Chaps. 2 and 3). It turns out that the phase space of a Hamiltonian system possesses certain geometric properties [1].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Henri Poincaré (1854–1912), a great French physicist and mathematician being i.a. a pioneer of chaos theory.
- 2.
Bernhard Riemann (1826–1866), influential German mathematician who made essential contributions to analysis and differential geometry.
- 3.
Paul Finsler (1894–1970), German and Swiss mathematician.
- 4.
We will often use both terms in an interchangeable way.
- 5.
We emphasize that it is not the only way to obtain the metric tensor.
- 6.
Tullio Levi-Civita (1873–1941), Italian mathematician of Jewish origin who investigated celestial mechanics, the three-body problem, and hydrodynamics.
- 7.
Obviously, we can obtain geodesic equations from formula (10.1.5).
- 8.
Elwin Bruno Christoffel (1829–1900), German mathematician and physicist working mainly at the University of Strasbourg.
- 9.
Since, by definition, a Riemannian space has the structure of a differentiable manifold, in general there is no single global coordinate system but many so-called local coordinate systems.
- 10.
Often this quantity is called the absolute derivative of a tensor [8].
- 11.
A similar situation occurs in the case of calculation of Lyapunov exponents.
- 12.
Obviously we apply here the Einstein summation convention, that is, we carry out the summation with respect to repeating indices.
References
V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978)
H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste (Gauthier-Villars, Paris, 1892) (in French)
J.V. Jose, E.J. Saletan, Classical Dynamics: A Contemporary Approach (Cambridge University Press, Cambridge, 1998)
L. Casetti, M. Pettini, E.D.G. Cohen, Geometric approach to hamiltonian dynamics and statistical mechanics. Phys. Rep. 337(3), 237–342 (2000)
P. Finsler, Über Kurven und Flachen in allegemeinen Raumen. Dissertation, Göttingen, 1918 (Birkhauser, Basel, 1951) (in German)
M.P. do Carmo, Riemannian Geometry (Birkhauser, Boston, 1993)
M. Nakahara, Geometry, Topology and Physics (Adam Hilger, Bristol, 1990)
L.U. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1925)
I.M. Gelfand, S.W. Fomin, Calculus of Variations (Prentice Hall, Englewood Cliffs, 1963)
J.L. Synge, A. Schild, Tensor Calculus (Dover, New York, 1978)
G. Białkowski, Classical Mechanics (PWN, Warsaw, 1975) (in Polish)
J. Awrejcewicz, Classical Mechanics: Statics and Kinematics (Springer, New York, 2012)
J. Awrejcewicz, Bifurcation and Chaos in Simple Dynamical Systems (World Scientific, Singapore, 1989)
J. Awrejcewicz, D. Sendkowski, M. Kazmierczak, Geometrical approach to the swinging pendulum dynamics. Comput. Struct. 84(24–25), 1577–1583 (2006)
J. Awrejcewicz, D. Sendkowski, Geometric analysis of the dynamics of a double pendulum. J. Mech. Mater. Struct. 2(8), 1421–1430 (2007)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Awrejcewicz, J. (2012). Geometric Dynamics. In: Classical Mechanics. Advances in Mechanics and Mathematics, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3740-6_11
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3740-6_11
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3739-0
Online ISBN: 978-1-4614-3740-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)