“Kolmogorov-Arnold-Moser Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Kolmogorov-Arnold-Moser Theorem

释义

Kolmogorov-Arnold-Moser Theorem

A theorem outlined in 1954 by Kolmogorov which was subsequently proved in the 1960s by Arnold and Moser (Tabor 1989,p. 105). It gives conditions under which Chaos is restricted in extent. Moser's 1962 proof was valid forTwist Maps

			(1)
			(2)

In 1963, Arnold produced a proof for Hamiltonian systems

(3)

The original theorem required perturbations

, although this has since been significantly increased. Arnold's proof required

, and Moser's original proof required

. Subsequently, Moser's version has beenreduced to

, then

, although counterexamples are known for

. Conditions for applicability of the KAMtheorem are:

: 1. small perturbations,
: 2. smooth perturbations, and
: 3. sufficiently irrational Winding Number.

Moser considered an integrable Hamiltonian function with a Torus and set of frequencies having anincommensurate frequency vector (i.e., for all Integers ). Let be perturbed by some periodic function . The KAM theorem states that, if is small enough, then foralmost every there exists an invariant Torus of the perturbed system such that is ``close to'' . Moreover, the Tori form a set of Positive measures whosecomplement has a measure which tends to zero as . A useful paraphrase of the KAM theorem is, ``For sufficientlysmall perturbation, almost all Tori (excluding those with rational frequency vectors) are preserved.'' Thetheorem thus explicitly excludes Tori with rationally related frequencies, that is, conditions of theform

(4)

These Tori are destroyed by the perturbation. For a system with two Degrees of Freedom, the condition of closed orbits is

(5)

For a Quasiperiodic Orbit,

is Irrational. KAM shows that the preserved Tori satisfy the irrationality condition

(6)

for all

and

, although not much is known about

The KAM theorem broke the deadlock of the small divisor problem in classical perturbation theory, and provides the startingpoint for an understanding of the appearance of Chaos. For a Hamiltonian System, the IsoenergeticNondegeneracy condition

(7)

guarantees preservation of most invariant Tori under small perturbations

. The Arnold versionstates that

(8)

for all

. This condition is less restrictive than Moser's, so fewer points are excluded.

See also Chaos, Hamiltonian System, Quasiperiodic Function, Torus

References

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

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