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The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, i.e. there is a map from the original manifold to the deformed one that is continuous in the perturbation. Conversely, other invariant tori are destroyed: even
172:
arbitrarily small perturbations cause the manifold to no longer be invariant and there exists no such map to nearby manifolds. Surviving tori meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies. This implies that the motion on the deformed torus continues to be
480:
866:{\displaystyle \exists ~\gamma ,\tau >0{\text{ such that }}|{\boldsymbol {\omega }}\cdot {\boldsymbol {k}}|\geq {\frac {\gamma }{\|{\boldsymbol {k}}\|^{\tau }}},\forall ~{\boldsymbol {k}}\in \mathbb {Z} ^{d}\backslash \left\{{\boldsymbol {0}}\right\}}
194:
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.
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The existence of a KAM theorem for perturbations of quantum many-body integrable systems is still an open question, although it is believed that arbitrarily small perturbations will destroy integrability in the infinite size limit.
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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.
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176:, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem quantifies the level of perturbation that can be applied for this to be true.
626:
926:
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509:
290:
952:
675:
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517:
384:
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1065:
V. I. Arnold, "Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the
Hamiltonian ,"
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218:
The methods introduced by
Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasiperiodic motions, now known as
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1336:
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639:
210:
An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.
222:. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of
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1234:
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475:{\displaystyle \mathbb {T} ^{d}:=\underbrace {\mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}} _{d}}
1331:
1302:
The KAM Story – A Friendly
Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory
223:
993:
1168:, 2nd ed., Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem. Springer 1997.
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A. N. Kolmogorov, "On the
Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian ,"
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in phase space. Plotting the coordinates of an integrable system would show that they are quasiperiodic.
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47:
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showed how to eliminate this degeneracy by developing a rotation-invariant version of the theorem.
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718:{\displaystyle {\boldsymbol {k}}\in \mathbb {Z} ^{d}\backslash \left\{{\boldsymbol {0}}\right\}}
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85:
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1230:
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case is normally excluded in classical KAM theory because it does not involve small divisors.
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144:
119:
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1126:
Percival, I C (1979-03-01). "A variational principle for invariant tori of fixed frequency".
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998:
73:
43:
555:{\displaystyle \mathrm {d} {\boldsymbol {\varphi }}/\mathrm {d} t={\boldsymbol {\omega }}}
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89:
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81:
1139:
226:) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).
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359:{\displaystyle {\boldsymbol {\varphi }}:{\mathcal {T}}^{d}\rightarrow \mathbb {T} ^{d}}
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Arnold originally thought that this theorem could apply to the motions of the
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59:
1048:
J. Moser, "On invariant curves of area-preserving mappings of an annulus,"
17:
152:
118:
in his formulation of the problem for larger numbers of bodies. Later,
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665:{\displaystyle {\boldsymbol {k}}\cdot {\boldsymbol {\omega }}\neq 0}
159:
of the integrable
Hamiltonian system will trace different invariant
179:
Those KAM tori that are destroyed by perturbation become invariant
160:
1098:"Addendum to Arnold Memorial Workshop: Khesin on Pinzari's talk"
135:
The KAM theorem is usually stated in terms of trajectories in
907:
847:
699:
330:
244:
50:
under small perturbations. The theorem partly resolves the
592:{\displaystyle {\boldsymbol {\omega }}\in \mathbb {R} ^{d}}
76:. The original breakthrough to this problem was given by
65:
The problem is whether or not a small perturbation of a
96:), and the general result is known as the KAM theorem.
728:
and "badly" approximated by rationals, typically in a
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in 1954. This was rigorously proved and extended by
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1286:KAM theory: the legacy of Kolmogorov’s 1954 paper
1079:, 9--36, doi:10.1070/RM1963v018n05ABEH004130 ).
1128:Journal of Physics A: Mathematical and General
8:
806:
797:
1205:Proceedings of Symposia in Pure Mathematics
1166:Mathematical Methods of Classical Mechanics
599:is a non-zero constant vector, called the
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312:-torus, if there exists a diffeomorphism
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110:, but it turned out to work only for the
1198:"A lecture on the classical KAM-theorem"
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621:{\displaystyle {\boldsymbol {\omega }}}
614:
570:
548:
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69:dynamical system results in a lasting
1305:, 2014, World Scientific Publishing,
265:invariant under the action of a flow
7:
921:{\displaystyle {\mathcal {T}}^{d}}
821:
739:
537:
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511:is uniform linear but not static,
482:such that the resulting motion on
258:{\displaystyle {\mathcal {T}}^{d}}
25:
1272:"Kolmogorov-Arnold-Moser Theorem"
1172:Wayne, C. Eugene (January 2008).
855:
707:
504:{\displaystyle \mathbb {T} ^{d}}
1174:"An Introduction to KAM Theory"
1291:Kolmogorov-Arnold-Moser theory
784:
766:
341:
131:Integrable Hamiltonian systems
1:
1337:Theorems in dynamical systems
1164:Arnold, Weinstein, Vogtmann.
1052:Göttingen Math.-Phys. Kl. II
994:Stability of the Solar System
155:-shaped surface). Different
27:Result in dynamical systems
1358:
1257:Rafael de la Llave (2001)
1148:10.1088/0305-4470/12/3/001
103:or other instances of the
1227:10.1090/pspum/069/1858551
1072:(1963) (English transl.:
636:incommensurable, that is
285:{\displaystyle \phi ^{t}}
46:about the persistence of
1342:Computer-assisted proofs
1260:A tutorial on KAM theory
632:rationally independent (
606:If the frequency vector
1316:Chapter 1: Introduction
1196:Jürgen Pöschel (2001).
1102:James Colliander's Blog
947:{\displaystyle d\geq 2}
292:is called an invariant
32:Kolmogorov–Arnold–Moser
1009:Hofstadter's butterfly
978:
948:
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891:
867:
719:
666:
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556:
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92:in 1963 (for analytic
1332:Hamiltonian mechanics
1014:Nekhoroshev estimates
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762: such that
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477:
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52:small-divisor problem
48:quasiperiodic motions
1092:(October 24, 2011),
1034:Dokl. Akad. Nauk SSR
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84:in 1962 (for smooth
1140:1979JPhA...12L..57P
977:{\displaystyle d=1}
877:then the invariant
143:. The motion of an
94:Hamiltonian systems
60:classical mechanics
56:perturbation theory
54:that arises in the
1269:Weisstein, Eric W.
1050:Nachr. Akad. Wiss.
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366:into the standard
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157:initial conditions
147:is confined to an
141:Hamiltonian system
112:three-body problem
1311:978-981-4556-58-3
1108:on March 29, 2017
1094:Colliander, James
1074:Russ. Math. Surv.
1067:Uspekhi Mat. Nauk
890:{\displaystyle d}
826:
816:
763:
744:
410:
408:
379:{\displaystyle d}
305:{\displaystyle d}
145:integrable system
139:of an integrable
120:Gabriella Pinzari
78:Andrey Kolmogorov
44:dynamical systems
16:(Redirected from
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1239:. Archived from
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1104:, archived from
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999:Arnold diffusion
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149:invariant torus
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90:Vladimir Arnold
42:is a result in
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1134:(3): L57–L60.
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1004:Ergodic theory
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224:Michael Herman
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1246:on 2016-03-03
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1090:Khesin, Boris
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1059:
1056:(1962), 1–20.
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174:quasiperiodic
167:Perturbations
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114:because of a
113:
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108:-body problem
102:
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71:quasiperiodic
68:
63:
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53:
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33:
19:
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1300:
1295:Scholarpedia
1275:
1258:
1248:. Retrieved
1241:the original
1208:
1204:
1185:. Retrieved
1180:
1165:
1131:
1127:
1121:
1110:, retrieved
1106:the original
1101:
1084:
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1073:
1069:
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1061:
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1044:
1036:
1033:
1028:
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876:
729:
633:
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512:
233:
219:
217:
209:
206:Consequences
200:
197:
193:
184:
178:
170:
134:
101:Solar System
98:
82:Jürgen Moser
67:conservative
64:
39:
35:
31:
29:
1211:: 707–732.
730:Diophantine
234:A manifold
181:Cantor sets
137:phase space
18:KAM theorem
1326:Categories
1250:2006-06-06
1159:References
220:KAM theory
214:KAM theory
116:degeneracy
86:twist maps
1277:MathWorld
1213:CiteSeerX
1112:March 29,
956:KAM torus
939:≥
848:∖
833:∈
822:∀
811:τ
807:‖
798:‖
794:γ
789:≥
776:⋅
772:ω
752:τ
746:γ
740:∃
700:∖
685:∈
657:≠
653:ω
649:⋅
615:ω
575:∈
571:ω
549:ω
528:φ
462:⏟
446:×
443:⋯
440:×
425:×
342:→
321:φ
274:ϕ
230:KAM torus
191:in 1979.
126:Statement
1181:Preprint
988:See also
732:sense:
672:for all
183:, named
153:doughnut
1187:20 June
1136:Bibcode
1096:(ed.),
1039:(1954).
897:-torus
562:，where
386:-torus
185:Cantori
40:theorem
1309:
1233:
1215:
958:. The
825:
743:
634:a.k.a.
88:) and
1293:from
1244:(PDF)
1201:(PDF)
1177:(PDF)
1020:Notes
628:is:
74:orbit
1307:ISBN
1231:ISBN
1189:2012
1183:: 29
1114:2017
1054:1962
755:>
513:i.e.
161:tori
30:The
1223:doi
1144:doi
603:.
187:by
151:(a
58:of
36:KAM
1328::
1313:.
1274:.
1229:.
1221:.
1209:69
1207:.
1203:.
1179:.
1142:.
1132:12
1130:.
1100:,
1077:18
1070:18
1037:98
406::=
62:.
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1280:.
1263:.
1253:.
1225::
1191:.
1150:.
1146::
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942:2
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860:}
856:0
852:{
843:d
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542:t
538:d
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492:T
468:d
456:1
451:S
435:1
430:S
420:1
415:S
401:d
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374:d
352:d
347:T
337:d
331:T
325::
300:d
278:t
251:d
245:T
106:n
34:(
20:)
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