Citation: George Contopoulos. A Review of the “Third” integral[J]. Mathematics in Engineering, 2020, 2(3): 472-511. doi: 10.3934/mine.2020022
[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 |