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Topic: KAM theorem

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	Philosophy of Statistical Mechanics
		Poincaré's theorem showed that any appropriately bounded system in which energy was conserved would of necessity, over an infinite time, return an infinite number of times to states arbitrarily close to the initial dynamical state in which the system was started.
		Loschmidt argued that the time irreversibility of thermodynamics was incompatible with the symmetry under time reversal of the classical dynamics assumed to govern the motion of the molecular constituents of the object.
		On the other hand another result of dynamical theory, the Kolmogorov-Arnold-Moser (KAM) theorem shows that more realistic models (say of molecules interacting by means of "soft" potentials) are likely not to obey ergodicity in a strict sense.
	plato.stanford.edu /entries/statphys-statmech (5119 words)

	Mathematics Dept. WWW SERVER in Rome, Italy (Site not responding. Last check: 2007-10-11)
		G. Gentile, V. Mastropietro: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics Rev.
		G. Gentile, V. Mastropietro: KAM theorem revisited Physica D N.90, 225-234 (1996).
		G. Gentile, V. Mastropietro: Tree expansion and multiscale analysis for KAM tori, Nonlinearity N.8, 1995, 1159-1178.
	www.mat.uniroma2.it /~mastropi/pubbing.html (813 words)

	Nonlinear Science FAQ (Site not responding. Last check: 2007-10-11)
		In fact the famous Poincaré-Bendixson theorem states that for a pair of first order differential equations, only fixed points and limit cycles can occur (there is no chaos in 2D flows).
		Mathematically this was shown to be the case by the celebrated theorem of Kolmogorov Arnold and Moser (KAM), first proposed by Kolmogorov in 1954.
		The KAM theorem is rather technical, but in essence says that many of the quasiperiodic motions are preserved under perturbations.
	www.lapasserelle.com /Nonlinear%20Science%20FAQ.htm (11615 words)

	Cogprints - Constructive Methods of Invariant Manifolds for Kinetic Problems (Site not responding. Last check: 2007-10-11)
		A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems.
		The systematic use of thermodynamics structures and of the quasi--chemical representation allow to construct approximations which are in concordance with physical restrictions.
		Arnold, V. I., Proof of a theorem of A N Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian.
	cogprints.org /3087 (3642 words)

	Research Output: Department of Mathematics (Site not responding. Last check: 2007-10-11)
		Georgiou [2003] KAM Theorem and Stability of the Upside-Down Pendulum.
		Melbourne and A. Török [2003] Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions.
		Melbourne and A. Török [2002] Central limit theorems and invariance principles for time-one maps of hyperbolic flows.
	www.maths.surrey.ac.uk /research/research-output.php (6758 words)

	Borsuk-Ulam theorem (Site not responding. Last check: 2007-10-11)
		Using the Borsuk-Ulam Theorem : Lectures on Topological Methods in Combinatorics...
		buch.de - bücher - Using the Borsuk-Ulam Theorem - Jiri Matousek...
		Geometry.Net - Theorems And Conjectures Books: Geometry Theorem...
	www.scienceoxygen.com /math/669.html (83 words)

	New Library Orders for 2003
		A key feature of the book is the early introduction of geometric (differential manifold) ideas, as well as detailed treatment of topics in nonlinear dynamics (such as the KAM theorem) and continuum dynamics (including solitons).
		And the mathematics, as it is done, is shaped and supported, or not, by convention, subject matter, calculation, and analogy.
		The cases studied include the central limit theorem of statistics, the sound of the shape of a drum, the connection between algebra and topology, the stability of matter, the Ising model, and the Langlands Program in number theory and representation theory.
	www.uwec.edu /physics/Library/2003-4_orders.htm (11704 words)

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