A Review of the “Third” integral

George Contopoulos; George Contopoulos

doi:10.3934/mine.2020022

Mathematics in Engineering

2020, Volume 2, Issue 3: 472-511. doi: 10.3934/mine.2020022

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Review Special Issues

A Review of the “Third” integral

George Contopoulos ^,

Research Center for Astronomy of the Academy of Athens Soranou Efesiou 4, Athens, GR- 11527, Greece

Received: 01 July 2019 Accepted: 14 October 2019 Published: 11 March 2020

We present the history of the third integral from a personal viewpoint. In particular, we mention the discovery of particular forms of the third integral, especially in resonant cases, the generation of chaos due to resonance overlap, the nonlinear theory of spiral density waves and applications in relativity and cosmology and in quantum mechanics. Finally we refer to some recent developments concerning the use of the third integral in finding chaotic orbits.
- integrals of motion,
- resonances,
- density waves,
- orbits in relativity and cosmology,
- classical and quantum chaos
Citation: George Contopoulos. A Review of the “Third” integral[J]. Mathematics in Engineering, 2020, 2(3): 472-511. doi: 10.3934/mine.2020022

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Abstract

We present the history of the third integral from a personal viewpoint. In particular, we mention the discovery of particular forms of the third integral, especially in resonant cases, the generation of chaos due to resonance overlap, the nonlinear theory of spiral density waves and applications in relativity and cosmology and in quantum mechanics. Finally we refer to some recent developments concerning the use of the third integral in finding chaotic orbits.

References

[1]	Arnold VI (1961) On the stability of the equilibrium of a Hamiltonian system of ordinary differential equations in a generic elliptic case. Doklady USSR 137: 255-257.
[2]	Arnold VI (1963) A proof of the A.N. Kolmogorov's theorem on the conservation of conditional-periodic motions in a small change of the Hamiltonian function. Uspehy Math Nauk 18: 13-40.
[3]	Arnold VI (1963) Small denominators and problems on the stability of motions in the classical and celestial mechanics. Uspehy Math Nauk 18: 91-192.
[4]	Arnold VI (1964) Instability of dynamical systems with several degrees of freedom. Sov Math Dokl 5: 581-585.
[5]	Barbanis B (1962) An application of the third integral in the velocity space. Z Astrophys 56: 56-67.
[6]	Belinskii VA, Khalatnikov IM (1969) On the nature of the singularities in the general solution of the gravitational equations. Sov Phys JETP 29: 911-917.
[7]	Birkhoff GD (1927) Dynamical Systems, Providence: American Mathematical Soc.
[8]	Bohm D (1952) A suggested interpretation of the quantum theory in terms of "hidden" variables I & II. Phys Rev 85: 166-179, 180-193. doi: 10.1103/PhysRev.85.166
[9]	Bohm D (1953) Proof that probability density approaches \|ψ\|2 in causal interpretation of the quantum theory. Phys Rev 89: 458-466.
[10]	Bohm D, Vigier JP (1954) Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys Rev 26: 208-216.
[11]	Bozis G (1967) The applicability of a new integral in the restricted three-body problem. I. Astron J 72: 380-385. doi: 10.1086/110236
[12]	Broucke R (1969) Stability of periodic orbits in the elliptic restricted three-body problem. Amer Inst Astronaut Aeronaut J 7: 1003-1009. doi: 10.2514/3.5267
[13]	Chandrasekhar S (1989) The two-centre problem in general relativity: The scattering of radiation by two extreme Reissner-Nordström black-holes. Proc Roy Soc London A 421: 227-258. doi: 10.1098/rspa.1989.0010
[14]	Chandrasekhar S, Contopoulos G (1967) On a post-Galilean transformation appropriate to the post-Newtonian theory of Einstein, Infeld and Hoffmann. Proc Roy Soc London A 298: 123-141.
[15]	Cherry TM (1924) Integrals of systems of ordinary differential equations. Math Proc Cambridge Phil Soc 22: 273-281. doi: 10.1017/S0305004100014195
[16]	Cherry TM (1924) Note on the employment of angular variables in Celestial Mechanics. Month Not. Roy Astron Soc 84: 729-731. doi: 10.1093/mnras/84.9.729
[17]	Cherry TM (1928) On the solution of Hamiltonian systems of differential equations in the neighbourhood of a singular point. Proc London Math Soc 27: 151-170.
[18]	Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52: 263-279. doi: 10.1016/0370-1573(79)90023-1
[19]	Contopoulos G (1957) On the relative motions of stars in a galaxy. Stockholm Obs Ann 19: 10.
[20]	Contopoulos G (1958) On the vertical motions of stars in a galaxy. Stockholm Obs Ann 20: 5.
[21]	Contopoulos G (1960) A third integral of motion in a galaxy. Z Astrophys 49: 273-291.
[22]	Contopoulos G (1963) On the existence of a third integral of motion. Astron J 68: 1-14. doi: 10.1086/108903
[23]	Contopoulos G (1965) Periodic and "tube" orbits. Astron J 70: 526-544. doi: 10.1086/109777
[24]	Contopoulos G (1966) Tables of the third integral. Astrophys J Suppl 13: 503-608. doi: 10.1086/190145
[25]	Contopoulos G (1966) Recent developments in stellar dynamics. IAU Symposium 25: 3-18.
[26]	Contopoulos G (1966) Resonance phenomena and the non-applicability of the "third integral". Bull Astron 2: 223-241.
[27]	Contopoulos G (1968) Resonant periodic orbits. Astrophys J 153: 83-94. doi: 10.1086/149638
[28]	Contopoulos G (1970) Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron J 75: 96-107.
[29]	Contopoulos G (1970) Gravitational theories of spiral structure. IAU Symposium 38: 303-316.
[30]	Contopoulos G (1973) The particle resonance in spiral galaxies. Nonlinear effects. Astrophys J 181: 657-684.
[31]	Contopoulos G (1975) Inner Lindblad resonance in galaxies-Nonlinear theory. I. Astrophys J 201: 566-584. doi: 10.1086/153922
[32]	Contopoulos G (1980) How far do bars extend. Astron Astrophys 81: 198-209.
[33]	Contopoulos G (1984) Theoretical periodic orbits in 3-dimensional Hamiltonians. Physica D 11: 179-192. doi: 10.1016/0167-2789(84)90442-1
[34]	Contopoulos G (1986) Qualitative changes in 3-dimensional dynamical systems. Astron Astrophys 161: 244-256.
[35]	Contopoulos G (1988) The 4: 1 resonance in barred galaxies. Astron Astrophys 201: 44-50.
[36]	Contopoulos G (1990) Periodic orbits and chaos around two black holes. Proc Roy Soc London A 431: 183-302. doi: 10.1098/rspa.1990.0126
[37]	Contopoulos G (1991) Periodic orbits and chaos around two fixed black holes. II. Proc Roy Soc London A 435: 551-562. doi: 10.1098/rspa.1991.0160
[38]	Contopoulos G (2002) Order and Chaos in Dynamical Astronomy, Springer Verlag.
[39]	Contopoulos G (2004) Adventures in Order and Chaos: A Scientific Autobiography, Kluwer.
[40]	Contopoulos G, Efthymiopoulos C (2008) Ordered and chaotic Bohmian trajectories. Cel Mech Dyn Astron 102: 219-239. doi: 10.1007/s10569-008-9127-8
[41]	Contopoulos G, Harsoula M (2015) Convergence regions of the Moser normal forms and the structure of chaos. J Phys A 48: 335101. doi: 10.1088/1751-8113/48/33/335101
[42]	Contopoulos G, Magnenat P (1985) Simple three-dimensional periodic orbits in a galactic-type potential. Cel Mech Dyn Astron 37: 387-414.
[43]	Contopoulos G, Moutsoulas M (1965) Resonance cases and small divisors in a third integral of motion. II. Astron J 70: 817-835. doi: 10.1086/109822
[44]	Contopoulos G, Moutsoulas M (1966) Resonance cases and small divisors in a third integral of motion. III. Astron J 71: 687-698. doi: 10.1086/110173
[45]	Contopoulos G, Papadaki H (1993) Newtonian and relativistic periodic orbits around two fixed black holes. Cel Mech Dyn Astron 55: 47-85. doi: 10.1007/BF00694394
[46]	Contopoulos G, Woltjer L (1964) The "third" integral in non-smooth potentials. Astrophys J 140: 1106-1119. doi: 10.1086/148009
[47]	Contopoulos G, Galgani L, Giorgilli A (1978) On the number of isolating integrals in Hamiltonian systems. Phys Rev A 18: 1183-1189. doi: 10.1103/PhysRevA.18.1183
[48]	Contopoulos G, Grammaticos B, Ramani A (1993) Painlevé analysis for the mixmaster universe model. J Phys A 26: 5795-5799. doi: 10.1088/0305-4470/26/21/018
[49]	Contopoulos G, Grammaticos B, Ramani A (1995) The last remake of the mixmaster universe model. J Phys A 28: 5313-5322. doi: 10.1088/0305-4470/28/18/020
[50]	Contopoulos G, Grousousakou E, Polymilis C (1996) Distribution of periodic orbits and the homoclinic tangle. Cel Mech Dyn Astron 64: 363-381. doi: 10.1007/BF00054553
[51]	Contopoulos G, Efthymiopoulos C, Giorgilli A (2003) Non-convergence of formal integrals of motion. J Phys A 36: 8639-8660. doi: 10.1088/0305-4470/36/32/306
[52]	Contopoulos G, Delis N, Efthymiopoulos C (2012) Order in de Broglie-Bohm quantum mechanics. J Phys A 45: 165301. doi: 10.1088/1751-8113/45/16/165301
[53]	Contopoulos G, Tzemos A, Efthymiopoulos C (2017) Partial integrability of 3d Bohmian trajectories. J Phys A 50: 195101. doi: 10.1088/1751-8121/aa685d
[54]	Cornish NJ, Levin JJ (1997) The mixmaster universe is chaotic. Phys Rev Lett 78: 998-1001. doi: 10.1103/PhysRevLett.78.998
[55]	Cushman R, Sniatycki J (1995) Local integrability of the mixmaster model. Rep Math Phys 36: 75-89. doi: 10.1016/0034-4877(96)82485-2
[56]	da Silva Ritter GI, de Almeida AMO, Douady R (1987) Analytical determination of unstable periodic orbits in area preserving maps. Physica D 29: 181-190. doi: 10.1016/0167-2789(87)90054-6
[57]	de Broglie L (1926) Sur la possibilité de relier les phenomènes d' interference et de diffraction a la théorie des quanta de lumière. C R Acad Sci Paris 183: 447-448.
[58]	de Broglie L (1926) Interference and corpuscular light. Nature 118: 441-442.
[59]	de Broglie L (1927) La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J Physique et Radium 8: 225-241. doi: 10.1051/jphysrad:0192700805022500
[60]	de Broglie L (1927) La structure atomique de la matière et du rayonnement et la méecanique ondulatoire. C R Acad Sci Paris 184: 273-274.
[61]	de Broglie L (1927c) Sur le role des ondes continues en mécanique ondulatoire. C R Acad Sci Paris 185: 380-382.
[62]	Delis N, Efthymiopoulos C, Contopoulos G (2012) Quantum vortices and trajectories in particle diffraction. Int J Bifurcation Chaos 22: 1250214. doi: 10.1142/S0218127412502148
[63]	Deprit A (1969) Canonical transformations depending on a small parameter. Cel Mech 1: 12-30. doi: 10.1007/BF01230629
[64]	Efthymiopoulos C (2005) Formal integrals and Nekhoroshev stability in a mapping model for the Trojan asteroids. Cel Mech Dyn Astron 92: 29-52. doi: 10.1007/s10569-004-4495-1
[65]	Efthymiopoulos C, Contopoulos G (2006) Chaos in Bohmian quantum mechanics. J Phys A 39: 1819-1852. doi: 10.1088/0305-4470/39/8/004
[66]	Efthymiopoulos C, Harsoula M (2013) The speed of Arnold diffusion. Phys D 251: 19-38. doi: 10.1016/j.physd.2013.01.016
[67]	Efthymiopoulos C, Sandor Z (2005) Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion. Month Not Roy Astron Soc 364: 253-271. doi: 10.1111/j.1365-2966.2005.09572.x
[68]	Efthymiopoulos C, Giorgilli A, Contopoulos G (2004) Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation. J Phys A 37: 10831-10858.
[69]	Efthymiopoulos C, Kalapotharakos C, Contopoulos G (2007) Nodal points and the transition from ordered to chaotic Bohmian trajectories. J Phys A 40: 12945-12971. doi: 10.1088/1751-8113/40/43/008
[70]	Efthymiopoulos C, Kalapotharakos C, Contopoulos G (2009) Origin of chaos near critical points of quantum flow. Phys Rev E 79: 036203. doi: 10.1103/PhysRevE.79.036203
[71]	Efthymiopoulos C, Delis N, Contopoulos G (2012) Wavepacket approach to particle diffraction by thin targets: Quantum trajectories and arrival times. Ann Phys 327: 438-460. doi: 10.1016/j.aop.2011.10.006
[72]	Efthymiopoulos C, Contopoulos G, Katsanikas M (2014) Analytical invariant manifolds near unstable points and the structure of chaos. Cel Mech Dyn Astron 119: 331-356. doi: 10.1007/s10569-014-9546-7
[73]	Fermi E (1923) Beweis dass ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist. Phys Zeitschrift 24: 261-265.
[74]	Fermi E (1924) Über die existenz quasi-ergodischer systeme. Phys Zeitschrift 25: 166-167.
[75]	Froeschlé C, Guzzo M, Lega E (2000) Graphical evolution of the Arnold web: From order to chaos. Science 289: 2108-2110. doi: 10.1126/science.289.5487.2108
[76]	Giorgilli A (1979) A computer program for integrals of motion. Comput Phys Commun 16: 331-343. doi: 10.1016/0010-4655(79)90040-7
[77]	Giorgilli A (1988) Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point. Ann Inst H Poincare 48: 423-439.
[78]	Giorgilli A (1990) Les Méthodes Modernes de la Mécanique Céleste, Gif-sur-Yvette.
[79]	Giorgilli A (2001) Unstable equilibria of Hamiltonian systems. Discrete Cont Dyn Syst 7: 855-872. doi: 10.3934/dcds.2001.7.855
[80]	Giorgilli A, Galgani L (1978) Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Cel Mech 17: 267-280. doi: 10.1007/BF01232832
[81]	Giorgilli A, Morbidelli A (1997) Invariant KAM tori and global stability for Hamiltonian systems. Z Angew Math Phys 48: 102-134. doi: 10.1007/PL00001462
[82]	Gustavson FG (1966) Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point. Astron J 71: 670-686. doi: 10.1086/110172
[83]	Harsoula M, Contopoulos G, Efthymiopoulos C (2015) Analytical description of the structure of chaos. J Phys A 48: 135102. doi: 10.1088/1751-8113/48/13/135102
[84]	Harsoula M, Efthymiopoulos C, Contopoulos G (2016) Analytical forms of chaotic spiral arms. Month Not Roy Astron Soc 459: 3419-3431. doi: 10.1093/mnras/stw748
[85]	Hénon M (1966) Exploration numerique du problème restreint IV, Masses egales, orbites non periodiques. Bull Astron 1: 49-66.
[86]	Hénon M (1966) Numerical exlporation of the restricted three-body problem. IAU Symposium 25: 157-163.
[87]	Hénon M, Heiles C (1964) The applicability of the third integral of motion: Some numerical experiments. Astron J 69: 73-79. doi: 10.1086/109234
[88]	Hietarinta J (1987) Direct methods for the search of the second invariant. Phys Rep 147: 87-154. doi: 10.1016/0370-1573(87)90089-5
[89]	Hori GI (1966) Theory of general perturbation with unspecified canonical variable. Publ Astron Soc JPN 18: 287-296.
[90]	Kaluza M, Robnik M (1992) Improved accuracy of the Birkhoff-Gustavson normal form and its convergence properties. J Phys A 25: 5311. doi: 10.1088/0305-4470/25/20/013
[91]	Kaufmann DE, Contopoulos G (1996) Self-consistent models of barred spiral galaxies. Astron Astrophys 309: 381-402.
[92]	Kolmogorov AN (1954) On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl Akad Nauk SSSR 98: 527-530.
[93]	Lakshamanan M, Sahadevan R (1993) Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type. Phys Rep 224: 1-93. doi: 10.1016/0370-1573(93)90081-N
[94]	Latifi A, Musette M, Conte R (1994) The Bianchi IX (mixmaster) cosmological model is not integrable. Phys Lett A194: 83-92.
[95]	Lega E, Guzzo M, Froeschlé C (2003) Detection of Arnold diffusion in Hamiltonian systems. Physica D 182: 179-187. doi: 10.1016/S0167-2789(03)00121-0
[96]	Lichtenberg A, Lieberman M (1992) Regular and Chaotic Dynamics, Springer Verlag.
[97]	Lin CC, Shu FH (1964) On the spiral structure of disk galaxies. Astrophys J 140: 646-655. doi: 10.1086/147955
[98]	Lindblad B (1941) On the development of spiral structure in a rotating stellar system. Stockholm Obs Ann 13: 10.
[99]	Lindblad B, Langebartel R (1953) On the dynamics of stellar systems. Stockholm Obs Ann 17: 6.
[100]	Lynden-Bell D (1962) Stellar dynamics: Exact solution of the self-gravitation equation. Month Not Roy Astron Soc 123: 447-458.
[101]	Misner CM (1969) Mixmaster universe. Phys Rev Lett 22: 1071. doi: 10.1103/PhysRevLett.22.1071
[102]	Morbidelli A, Giorgilli A (1995) Superexponential stability of KAM tori. J Stat Phys 78: 1607-1617. doi: 10.1007/BF02180145
[103]	Moser J (1956) The analytic invariants of an area-preserving mapping near a hyperbolic fixed point. Commun Pure Appl Math 9: 673-692. doi: 10.1002/cpa.3160090404
[104]	Moser J (1958) New aspects in the theory of stability of Hamiltonian systems. Commun Pure Appl Math 11: 81-114. doi: 10.1002/cpa.3160110105
[105]	Moser J (1962) On invariant curves of area-preserving mapping of an annulus. Nachr Acad Wiss Göttingen II: 1-20.
[106]	Moser J (1967) Convergent series expansions for quasi-periodic motions. Math Ann 169: 136-176. doi: 10.1007/BF01399536
[107]	Moser J (1968) Lectures on Hamiltonian systems. Mem Amer Math Soc 81: 1-60.
[108]	Nekhoroshev NN (1977) An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russ Math Surv 32: 5-66.
[109]	Ollongren A (1962) Three-dimensional galactic stellar orbits. Bull Astron Neth 16: 241-296.
[110]	de Almeida AMO, Vieira WM (1997) Extended convergence of normal forms around unstable equilibria. Phys Lett A 227: 298-300. doi: 10.1016/S0375-9601(97)00037-6
[111]	Poincaré H (1892) Les Méthodes Nouvelles de la Mécanique Céleste, Paris: Gauthier Villars.
[112]	Ramani A, Grammaticos B, Bountis T (1989) The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys Rep 180: 159-245. doi: 10.1016/0370-1573(89)90024-0
[113]	Rosenbluth MN, Sagdeev RA, Taylor JB, et al. (1966) Destruction of magnetic surfaces by magnetic field irregularities. Nucl Fusion 6: 253-266.
[114]	Siegel CL (1956) Vorlesungen über Himmelsmechanik, Springer Verlag.
[115]	Simó C, Vieiro A (2011) Some remarks on the abundance of stable periodic orbits inside homoclinic lobes. Physica D 240: 1936-1953. doi: 10.1016/j.physd.2011.09.007
[116]	Stäckel P (1890) Eine charackteristische eigenschaft der Flächen, deren linienelement gegeben wird. Math Ann 35: 91-103.
[117]	Stäckel P (1893) Über die Bewegung eines Punktes in einer n-fachen Mannigfaltigkeit. Math Ann 42: 537-563. doi: 10.1007/BF01447379
[118]	Szebehely V (1966) Numerical explorations of the restricted three-body problem. IAU Symposium 25: 163-169.
[119]	Torgard I, Ollongren A (1960) Nuffic International Summer Course in Science, Part X.
[120]	Tsigaridi L, Patsis PA (2013) The backbones of stellar structures in barred-spiral models-the concerted action of various dynamical mechanisms on galactic discs. Month Not Roy Astron Soc 434: 2922-2939. doi: 10.1093/mnras/stt1207
[121]	Tzemos A, Contopoulos G (2018) Integrals of motion in 3D Bohmian trajectories. J Phys A 51: 075101. doi: 10.1088/1751-8121/aaa092
[122]	Tzemos A, Contopoulos G, Efthymiopoulos C (2016) Origin of chaos in 3-d Bohmian trajectories. Phys Lett A 380: 3796-3802. doi: 10.1016/j.physleta.2016.09.016
[123]	Tzemos A, Efthymiopoulos C, Contopoulos G (2018) Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory. Phys Rev E 97: 042201. doi: 10.1103/PhysRevE.97.042201
[124]	Vieira WM, de Almeida AMO (1996) Study of chaos in hamiltonian systems via convergent normal forms. Physica D 90: 9-30. doi: 10.1016/0167-2789(95)00233-2
[125]	Voglis N, Contopoulos G (1994) Invariant spectra of orbits in dynamical systems. J Phys A 27: 4899-4909. doi: 10.1088/0305-4470/27/14/017
[126]	Whittaker ET (1916) On the Adelphic integral of the differential equations of dynamics. Proc Roy Soc Edinburgh 37: 95-116.
[127]	Whittaker ET (1937) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4 Eds., Cambridge University Press.
[128]	Zaslavsky GM, Chirikov BV (1972) Stochastic instability of non-linear oscillations. Sov Phys Uspekhi 14: 549-568. doi: 10.1070/PU1972v014n05ABEH004669

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