# Difference between revisions of "Quasi-periodic motion"

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− | < | + | A motion of a particle according to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766402.png" /> is a (vector-valued) quasi-periodic function. |

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− | is a (vector-valued) quasi-periodic function. | ||

====Comments==== | ====Comments==== | ||

− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766403.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766404.png" /> real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766405.png" />, that are incommensurate, i.e. independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766406.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766408.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q0766409.png" />. Then the "rectilinear motion" on the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664010.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664011.png" /> is quasi-periodic (and not periodic because of the incommensurability). This is typical in the sense that any quasi-periodic motion is a composition of this with a vector-valued periodic function. |

− | be | ||

− | real numbers, | ||

− | that are incommensurate, i.e. independent over | ||

− | i.e. | ||

− | |||

− | implies | ||

− | Then the "rectilinear motion" on the torus | ||

− | defined by | ||

− | is quasi-periodic (and not periodic because of the incommensurability). This is typical in the sense that any quasi-periodic motion is a composition of this with a vector-valued periodic function. | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | Let a Hamiltonian function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664012.png" /> define a completely-integrable system (cf. [[Completely-integrable differential equation|Completely-integrable differential equation]]). Then there are new symplectic coordinates, called action angle coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664013.png" /> in which the Hamiltonian takes the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664014.png" /> and the Hamiltonian equations become | |

− | |||

− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table> | |

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− | + | The phase space in this case fibres into invariant tori of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664016.png" />, and the motion on each is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664017.png" /> i.e. quasi-periodic (or periodic, or a mixture of these two, Arnol'd's theorem). (In [[#References|[a1]]] these motions are called conditionally periodic.) These tori are the level sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664019.png" />. Such a torus is called a non-resonant torus if the corresponding frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664020.png" /> are functionally independent, i.e. if | |

− | |||

− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table> | |

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− | |||

It can be shown that the non-resonant tori form a set of full measure. Nevertheless, invariant resonant tori do exist and they are mixed in such a way that they too form a dense set. | It can be shown that the non-resonant tori form a set of full measure. Nevertheless, invariant resonant tori do exist and they are mixed in such a way that they too form a dense set. | ||

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Now consider a small perturbation of an integrable system satisfying (a2): | Now consider a small perturbation of an integrable system satisfying (a2): | ||

− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664022.png" /></td> </tr></table> | |

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− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664023.png" /> is periodic in the angle variables. The famous KAM theorem (Kolmogorov–Arnol'd–Moser theorem) says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664024.png" /> is small enough, then for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664025.png" /> there exists an invariant torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664026.png" /> of the perturbed system that is close to the invariant torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664027.png" /> of the unperturbed system. The theorem was stated (in an analytic case) by A.N. Kolmogorov [[#References|[a2]]] and proved by V.I. Arnol'd and J. Moser (different cases) [[#References|[a3]]], [[#References|[a4]]]. Moreover, the union of the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664028.png" /> forms a set of positive measure and the measure of the complement goes to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664029.png" />. There is much more to this theorem. In particular, it specifies a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664030.png" /> which do indeed have invariant tori associated to them. These are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664031.png" /> which are "sufficiently incommensurate" , which means that they satisfy an equality |

− | is periodic in the angle variables. The famous KAM theorem (Kolmogorov–Arnol'd–Moser theorem) says that if | ||

− | is small enough, then for almost-all | ||

− | there exists an invariant torus | ||

− | of the perturbed system that is close to the invariant torus | ||

− | of the unperturbed system. The theorem was stated (in an analytic case) by A.N. Kolmogorov [[#References|[a2]]] and proved by V.I. Arnol'd and J. Moser (different cases) [[#References|[a3]]], [[#References|[a4]]]. Moreover, the union of the invariants | ||

− | forms a set of positive measure and the measure of the complement goes to zero as | ||

− | There is much more to this theorem. In particular, it specifies a set of | ||

− | which do indeed have invariant tori associated to them. These are | ||

− | which are "sufficiently incommensurate" , which means that they satisfy an equality | ||

− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664032.png" /></td> </tr></table> | |

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− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664033.png" /></td> </tr></table> | |

− | 0 | ||

− | |||

− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664034.png" />. There are "many" such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664035.png" />, enough for the theorem as stated. It is also true that some such condition is necessary. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664037.png" /> is rational, then there are arbitrarily-small perturbations which will destroy the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664038.png" /> (a fact already known to H. Poincaré). There are many other aspects to KAM theory, for instance KAM-type theorems for the case of equilibrium points and for systems with additional time-periodic forces acting on them, cf. e.g. [[#References|[a5]]], [[#References|[a1]]]. There are also converse KAM-theorems, which give conditions on a region of space which guarantee that there are no invariant tori, [[#References|[a11]]], [[#References|[a12]]]. Some of the invariant KAM tori of the system break up (as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664039.png" /> grows) into Cantor set like sets; these are in this context sometimes known as Cantori. |

− | There are "many" such | ||

− | enough for the theorem as stated. It is also true that some such condition is necessary. For example, if | ||

− | and | ||

− | is rational, then there are arbitrarily-small perturbations which will destroy the torus | ||

− | a fact already known to H. Poincaré). There are many other aspects to KAM theory, for instance KAM-type theorems for the case of equilibrium points and for systems with additional time-periodic forces acting on them, cf. e.g. [[#References|[a5]]], [[#References|[a1]]]. There are also converse KAM-theorems, which give conditions on a region of space which guarantee that there are no invariant tori, [[#References|[a11]]], [[#References|[a12]]]. Some of the invariant KAM tori of the system break up (as | ||

− | grows) into Cantor set like sets; these are in this context sometimes known as Cantori. | ||

− | If | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664040.png" />, the iso-energy surfaces are of dimension 3, the invariant KAM tori in them are of dimension 2 and these hence separate the energy surfaces into disjoint pieces. As a result the orbits which are not on the invariant tori are trapped between them. If the frequencies of the tori depend differentiably on the values of the action variables defining it, which is guaranteed by the iso-energy non-degeneracy condition |

− | the iso-energy surfaces are of dimension 3, the invariant KAM tori in them are of dimension 2 and these hence separate the energy surfaces into disjoint pieces. As a result the orbits which are not on the invariant tori are trapped between them. If the frequencies of the tori depend differentiably on the values of the action variables defining it, which is guaranteed by the iso-energy non-degeneracy condition | ||

− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664041.png" /></td> </tr></table> | |

− | |||

then it follows that the action variables for all initial values remain near there and exhibit no secular perturbation phenomena. For more than 2 degrees of freedom the invariant tori do not separate the iso-energy surfaces into disjoint pieces and it is indeed possible, also in the iso-energy non-degenerate case, that the action variables drift away from their initial values. This phenomenon is known as Arnol'd diffusion. | then it follows that the action variables for all initial values remain near there and exhibit no secular perturbation phenomena. For more than 2 degrees of freedom the invariant tori do not separate the iso-energy surfaces into disjoint pieces and it is indeed possible, also in the iso-energy non-degenerate case, that the action variables drift away from their initial values. This phenomenon is known as Arnol'd diffusion. | ||

− | An example of an integrable system is the planetary system with the interactions between the planets disregarded; adding this interaction gives the small perturbation | + | An example of an integrable system is the planetary system with the interactions between the planets disregarded; adding this interaction gives the small perturbation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076640/q07664042.png" />. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.N. Kolmogorov, "On the preservation of quasi-periodic motions in classical and celestial mechanics" ''Dokl. Akad. Nauk SSSR'' , '''98''' (1954) pp. 525–530 (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "Small denominators and the problem of stability of motion in classical and celestial mechanics" ''Russian Math. Surveys'' , '''18''' : 6 (1963) pp. 86–191 ''Uspekhi Mat. Nauk'' , '''18''' : 6 (1963) pp. 91–192</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Moser, "On invariant curves of area preserving mappings on an annulus" ''Nachr. Akad. Wiss. Göttingen Math. Phys.'' , '''K1''' (1962) pp. 1–20</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Tabor, "Chaos and integrability in nonlinear dynamics" , Wiley (Interscience) (1989) pp. Chapt. 3</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Gallovotti, "The elements of mechanics" , Springer (1983) pp. 466ff</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.J. Lichtenberg, M.A. Lieberman, "Regular and stochastic motion" , Springer (1983) pp. 159ff</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Moser, "Stable and random motions in dynamical systems" , Princeton Univ. Press (1973)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> W. Thirring, "Lehrbuch der Mathematischen Physik" , '''1. Klassische Dynamische Systeme''' , Springer (1977) pp. 138</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S. Sternberg, "Celestial mechanics" , '''I-II''' , Benjamin (1969) pp. Chapt. III, §11–12</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Knauf, "Closed orbits and converse KAM theory" ''Nonlinearity'' , '''3''' (1990) pp. 961–973</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> R.S. MacKay, J.D. Meiss, J. Strak, "Converse KAM theory for symplectic twist maps" ''Nonlinearity'' , '''2''' (1989) pp. 555–570</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.N. Kolmogorov, "On the preservation of quasi-periodic motions in classical and celestial mechanics" ''Dokl. Akad. Nauk SSSR'' , '''98''' (1954) pp. 525–530 (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "Small denominators and the problem of stability of motion in classical and celestial mechanics" ''Russian Math. Surveys'' , '''18''' : 6 (1963) pp. 86–191 ''Uspekhi Mat. Nauk'' , '''18''' : 6 (1963) pp. 91–192</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Moser, "On invariant curves of area preserving mappings on an annulus" ''Nachr. Akad. Wiss. Göttingen Math. Phys.'' , '''K1''' (1962) pp. 1–20</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Tabor, "Chaos and integrability in nonlinear dynamics" , Wiley (Interscience) (1989) pp. Chapt. 3</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Gallovotti, "The elements of mechanics" , Springer (1983) pp. 466ff</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.J. Lichtenberg, M.A. Lieberman, "Regular and stochastic motion" , Springer (1983) pp. 159ff</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Moser, "Stable and random motions in dynamical systems" , Princeton Univ. Press (1973)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> W. Thirring, "Lehrbuch der Mathematischen Physik" , '''1. Klassische Dynamische Systeme''' , Springer (1977) pp. 138</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S. Sternberg, "Celestial mechanics" , '''I-II''' , Benjamin (1969) pp. Chapt. III, §11–12</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Knauf, "Closed orbits and converse KAM theory" ''Nonlinearity'' , '''3''' (1990) pp. 961–973</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> R.S. MacKay, J.D. Meiss, J. Strak, "Converse KAM theory for symplectic twist maps" ''Nonlinearity'' , '''2''' (1989) pp. 555–570</TD></TR></table> |

## Revision as of 14:53, 7 June 2020

A motion of a particle according to , where is a (vector-valued) quasi-periodic function.

#### Comments

Let be real numbers, , that are incommensurate, i.e. independent over , i.e. , , implies . Then the "rectilinear motion" on the torus defined by is quasi-periodic (and not periodic because of the incommensurability). This is typical in the sense that any quasi-periodic motion is a composition of this with a vector-valued periodic function.

Let a Hamiltonian function define a completely-integrable system (cf. Completely-integrable differential equation). Then there are new symplectic coordinates, called action angle coordinates, in which the Hamiltonian takes the form and the Hamiltonian equations become

(a1) |

The phase space in this case fibres into invariant tori of dimension , and the motion on each is of the form i.e. quasi-periodic (or periodic, or a mixture of these two, Arnol'd's theorem). (In [a1] these motions are called conditionally periodic.) These tori are the level sets , . Such a torus is called a non-resonant torus if the corresponding frequencies are functionally independent, i.e. if

(a2) |

It can be shown that the non-resonant tori form a set of full measure. Nevertheless, invariant resonant tori do exist and they are mixed in such a way that they too form a dense set.

Now consider a small perturbation of an integrable system satisfying (a2):

where is periodic in the angle variables. The famous KAM theorem (Kolmogorov–Arnol'd–Moser theorem) says that if is small enough, then for almost-all there exists an invariant torus of the perturbed system that is close to the invariant torus of the unperturbed system. The theorem was stated (in an analytic case) by A.N. Kolmogorov [a2] and proved by V.I. Arnol'd and J. Moser (different cases) [a3], [a4]. Moreover, the union of the invariants forms a set of positive measure and the measure of the complement goes to zero as . There is much more to this theorem. In particular, it specifies a set of which do indeed have invariant tori associated to them. These are which are "sufficiently incommensurate" , which means that they satisfy an equality

where . There are "many" such , enough for the theorem as stated. It is also true that some such condition is necessary. For example, if and is rational, then there are arbitrarily-small perturbations which will destroy the torus (a fact already known to H. Poincaré). There are many other aspects to KAM theory, for instance KAM-type theorems for the case of equilibrium points and for systems with additional time-periodic forces acting on them, cf. e.g. [a5], [a1]. There are also converse KAM-theorems, which give conditions on a region of space which guarantee that there are no invariant tori, [a11], [a12]. Some of the invariant KAM tori of the system break up (as grows) into Cantor set like sets; these are in this context sometimes known as Cantori.

If , the iso-energy surfaces are of dimension 3, the invariant KAM tori in them are of dimension 2 and these hence separate the energy surfaces into disjoint pieces. As a result the orbits which are not on the invariant tori are trapped between them. If the frequencies of the tori depend differentiably on the values of the action variables defining it, which is guaranteed by the iso-energy non-degeneracy condition

then it follows that the action variables for all initial values remain near there and exhibit no secular perturbation phenomena. For more than 2 degrees of freedom the invariant tori do not separate the iso-energy surfaces into disjoint pieces and it is indeed possible, also in the iso-energy non-degenerate case, that the action variables drift away from their initial values. This phenomenon is known as Arnol'd diffusion.

An example of an integrable system is the planetary system with the interactions between the planets disregarded; adding this interaction gives the small perturbation .

#### References

[a1] | V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian) |

[a2] | A.N. Kolmogorov, "On the preservation of quasi-periodic motions in classical and celestial mechanics" Dokl. Akad. Nauk SSSR , 98 (1954) pp. 525–530 (In Russian) |

[a3] | V.I. Arnol'd, "Small denominators and the problem of stability of motion in classical and celestial mechanics" Russian Math. Surveys , 18 : 6 (1963) pp. 86–191 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 91–192 |

[a4] | J. Moser, "On invariant curves of area preserving mappings on an annulus" Nachr. Akad. Wiss. Göttingen Math. Phys. , K1 (1962) pp. 1–20 |

[a5] | M. Tabor, "Chaos and integrability in nonlinear dynamics" , Wiley (Interscience) (1989) pp. Chapt. 3 |

[a6] | G. Gallovotti, "The elements of mechanics" , Springer (1983) pp. 466ff |

[a7] | A.J. Lichtenberg, M.A. Lieberman, "Regular and stochastic motion" , Springer (1983) pp. 159ff |

[a8] | J. Moser, "Stable and random motions in dynamical systems" , Princeton Univ. Press (1973) |

[a9] | W. Thirring, "Lehrbuch der Mathematischen Physik" , 1. Klassische Dynamische Systeme , Springer (1977) pp. 138 |

[a10] | S. Sternberg, "Celestial mechanics" , I-II , Benjamin (1969) pp. Chapt. III, §11–12 |

[a11] | A. Knauf, "Closed orbits and converse KAM theory" Nonlinearity , 3 (1990) pp. 961–973 |

[a12] | R.S. MacKay, J.D. Meiss, J. Strak, "Converse KAM theory for symplectic twist maps" Nonlinearity , 2 (1989) pp. 555–570 |

**How to Cite This Entry:**

Quasi-periodic motion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quasi-periodic_motion&oldid=48392