Kolmogorov–Arnold–Moser theorem
Encyclopedia
The Kolmogorov–Arnold–Moser theorem is a result in dynamical system
s about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory
of classical mechanics
.
The problem is whether or not a small perturbation of a conservative
dynamical system results in a lasting quasiperiodic
orbit
. The original breakthrough to this problem was given by Andrey Kolmogorov
in 1954. This was rigorously proved and extended by Vladimir Arnold
(in 1963 for analytic Hamiltonian system
s) and Jürgen Moser
(in 1962 for smooth twist maps), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be directly as a whole applied to the motions of the solar system
, but is useful in generating corrections of astronomical models for proving of long-termed stability and of avoiding of orbital resonance
in solar system, although Arnold used the methods of KAM to prove the stability of elliptical orbits in the planar three-body problem
.
The KAM theorem is usually stated in terms of trajectories in phase space
of an integrable
Hamiltonian system
.
The motion of an integrable system is confined to a doughnut
-shaped surface, an invariant torus. Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting any of the coordinates of an integrable system would show that they are quasi-periodic.
The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, while others are destroyed. The ones that survive are those that have “sufficiently irrational” frequencies (this is known as the non-resonance condition). This implies that the motion continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true. An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.
The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of Michael Herman) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).
The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. As the number of dimensions of the system increases, the volume occupied by the tori decreases.
Those KAM tori that are not destroyed by perturbation become invariant Cantor set
s, named Cantori by Ian C. Percival in 1979.
As the perturbation increases and the smooth curves disintegrate we move from KAM theory to
Aubry-Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
of classical mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
.
The problem is whether or not a small perturbation of a conservative
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...
dynamical system results in a lasting quasiperiodic
Quasiperiodic motion
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number of incommensurable frequencies....
orbit
Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
. The original breakthrough to this problem was given by Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
in 1954. This was rigorously proved and extended by Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
(in 1963 for analytic Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s) and Jürgen Moser
Jürgen Moser
Jürgen Kurt Moser or Juergen Kurt Moser was a German-American mathematician.-Professional biography:...
(in 1962 for smooth twist maps), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be directly as a whole applied to the motions of the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
, but is useful in generating corrections of astronomical models for proving of long-termed stability and of avoiding of orbital resonance
Orbital resonance
In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of...
in solar system, although Arnold used the methods of KAM to prove the stability of elliptical orbits in the planar three-body problem
Three-body problem
Three-body problem has two distinguishable meanings in physics and classical mechanics:# In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then...
.
The KAM theorem is usually stated in terms of trajectories in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of an integrable
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
Hamiltonian system
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
.
The motion of an integrable system is confined to a doughnut
Doughnut
A doughnut or donut is a fried dough food and is popular in many countries and prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franchised specialty outlets...
-shaped surface, an invariant torus. Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting any of the coordinates of an integrable system would show that they are quasi-periodic.
The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, while others are destroyed. The ones that survive are those that have “sufficiently irrational” frequencies (this is known as the non-resonance condition). This implies that the motion continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true. An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.
The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of Michael Herman) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).
The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. As the number of dimensions of the system increases, the volume occupied by the tori decreases.
Those KAM tori that are not destroyed by perturbation become invariant Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
s, named Cantori by Ian C. Percival in 1979.
As the perturbation increases and the smooth curves disintegrate we move from KAM theory to
Aubry-Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.