Skip to main content

The Adiabatic Invariant in Classical Mechanics

  • Chapter
Dynamics Reported

Part of the book series: Dynamics Reported ((DYNAMICS,volume 2))

Abstract

The adiabatic invariant theory in Classical Mechanics emerged from a very rich, but somewhat murky, sea of analogies with other types of problems in Theoretical Physics. We recall in the introduction of the first part how it derives from the “adiabatic principle” of Ehrenfest (1916) in the frame of the old quantum theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aamodt, R.E., 1971, Particle containment in mirror traps in the presence of fluctuating electric fields, Phys. Rev. Lett. 27, 135–138

    Article  Google Scholar 

  • Aamodt, R.E., 1972, Mirror containment with low-frequency short-wavelength fluctuations, Phys. of Fluids 15, 512–514

    Article  Google Scholar 

  • Abramowitz M. and Stegun, I.A., 1965, Handbook of Mathematical functions, Dover P.I., New York

    Google Scholar 

  • Alfven, H., 1950, Cosmical Electrodynamics, Clarendon Press, Oxford

    MATH  Google Scholar 

  • Allan, R.R., 1969, Evolution of mimas-tethys commensurability, Astron. J. 74, 497–508

    Article  Google Scholar 

  • Allan, R.R., 1970, On the evolution of commensurabilities between natural satellites, Symposia Mathematica 3, 75–96

    Google Scholar 

  • Andronov, A.A., Vitt, A.A. and Khaikin, S.E., 1966, Theory of Oscillators, Addison-Wesley, Reading, Mass.

    MATH  Google Scholar 

  • Arnol’d, V.I., 1964, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Survey 18, 85–191

    Article  Google Scholar 

  • Arnol’d, V.I., 1978, Mathematical Methods of Classical Mechanics, Springer-Verlag

    Google Scholar 

  • Arnol’d, V.I., 1980, Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordinaires, Mir, Moscou)

    MATH  Google Scholar 

  • Arnol’d, V.l. and Avez, A., 1968, Ergodic Problems of Classical Mechanics, W.A. Benjamin Inc.

    Google Scholar 

  • Best, R.W.B., 1968, On the motion of charged particles in a slightly damped sinusoidal potential wave, Physica 40, 182–196

    Article  Google Scholar 

  • Birkhoff, G., 1927, Dynamical Systems, Am. Math. Soc. Coll. Pub. IX

    Google Scholar 

  • Bogoliubov, N.M. and Mitropolsky, 1961, Asymptotic Methods in the Theory of Nonlinear Oscillations, Hindustan P.C.

    Google Scholar 

  • Borderies, N., 1980, La Rotation de Mars. Thesis, University of Toulouse Borderies, N. and Goldreich, P., 1984, A simple derivation of capture probabilities for the j + 1/j andj + 2/j orbit-orbit resonance problems, Celestial Mechanics 32, 127–136

    Google Scholar 

  • Born, H. and Fock, V., 1928, Beweis des Adiabatensatzes, Z. Physik 51, 165–180

    Article  Google Scholar 

  • Bretagnon, P., 1974, Termes à longues périodes dans le système solaire, Astron. Astro-phys. 30, 141–154

    MATH  Google Scholar 

  • Brouwer, D., 1959, Solution of the problem of an artificial satellite without drag, Astron. J. 64, 378

    Google Scholar 

  • Brouwer, D. and Clemence, G.M., 1961, Methods of Celestial Mechanics, Academic Press, N.Y.

    Google Scholar 

  • Burns, T.J., 1979, On the rotation of Mercury, Celestial Mechanics 19, 297–313

    Article  MathSciNet  MATH  Google Scholar 

  • Cary, J.R., Escande, D.F. and Tennyson, J.L., 1984, Institute for Fusion Studies, Report no. IFRS-155

    Google Scholar 

  • Cary, J.R., Escande, D.F. and Tennyson, J.L., 1986, Adiabatic invariant change due to separatrix crossing, Physical Review A34, 4256–4275

    Google Scholar 

  • Cassini, G.D., 1693, Traité de l’Origine et des Progrès de l’Astronomie, Paris

    Google Scholar 

  • Chirikov, B.V., 1959, The passage of a nonlinear oscillating system through resonance, Sov. Phys. Doklady 4, 390–394

    MathSciNet  MATH  Google Scholar 

  • Chirikov, B.V., 1979, A universal instability of many-dimensional oscillator systems, Physics Reports 52, 263–379

    Article  MathSciNet  Google Scholar 

  • Colombo, G., 1965, Rotation period of the planet Mercury, Nature 208, 575

    Article  Google Scholar 

  • Colombo, G., 1966, Cassini’s second and third laws, Astron. J. 71, 891–896

    Article  Google Scholar 

  • Colombo, G. and Shapiro, I.I., 1966, The rotation of the planet Mercury, Astrophys. J. 145, 296–307

    Article  Google Scholar 

  • Counselman C.C. and Shapiro, I.I., 1970, Spin-orbit resonance of Mercury, Symposia Mathematica, 3, 121–169

    Google Scholar 

  • Deprit, A., 1967, Free rotation of a ridig body studied in the phase plane, Am. J. of Phys. 35, 424–428

    Article  Google Scholar 

  • Deprit, A., 1969, Canonical transformations depending on a small parameter, Celestial Mechanics 1, 12–30

    Article  MathSciNet  MATH  Google Scholar 

  • Deprit, A. and Henrard, J., 1969, Construction of orbits asymptotic to a periodic orbit, Astron. J. 74, 308–318

    Article  Google Scholar 

  • Delaunay, C., 1867, Théorie du mouvement de la Lune, Mém. Acad. Sci. Paris 29

    Google Scholar 

  • Dobrott, D. and Greene, J.M., 1971, Probability of trapping-state transitions in a toroidal device, Phys. of Fluids 14, 1525–1531

    Article  Google Scholar 

  • Dommanget, J., 1963, Recherches sur l’évolution des étoiles doubles, Ann. Observatoire Royal de Belgique, III, 9, 7–92

    Google Scholar 

  • Ehrenfest, P., 1916, Adiabatische Invarianten und Quantentheorie, Ann. d. Phys. 51, 327

    Google Scholar 

  • Escande, D.F., 1985, Change of adiabatic invariant at separatrix crossing: Application to slow Hamiltonian chaos, in “Advances in Nonlinear Dynamics and Stochastic Processes” (R. Livi and A. Politi eds.), World Scientific Singapore, 67–79

    Google Scholar 

  • Escande, D.F., 1985, Stochasticity in classical Hamiltonian systems: universal aspects, Physics Reports 121, 165–261

    MathSciNet  Google Scholar 

  • Escande, D.F., 1987a, Slow Hamiltonian Chaos in “Advances in Nonlinear Dynamics and Stochastic Processes II” (Paladin and Vulpiani eds). World Scientific Singapore

    Google Scholar 

  • Escande, D.F., 1987b., Hamiltonian Chaos and adiabaticity. Proc. Int. Workshop Kiev. World Scientific

    Google Scholar 

  • Freidberg, J.P., 1982, Ideal magnetohydrodynamic theory of magnetic fusion systems, Rev. of Modern Physics 54, 801–902

    Article  Google Scholar 

  • Gardner, C.S., 1959, Adiabatic invariants of periodic classical systems, Phys. Rev. 115, 791–794

    Article  MathSciNet  Google Scholar 

  • Giacaglia, G.E.O., 1972, Perturbation Methods in Non-Linear Systems, Springer-Verlag

    MATH  Google Scholar 

  • Giorgilli, A. and Galgani, L., 1985, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celestial Mechanics 37, 95–112

    Article  MathSciNet  Google Scholar 

  • Goldreich, P., 1965, An explanation of the frequent occurrence of commensurable mean motions in the Solar System, M.N.R.A.S. 130, 159–181

    Google Scholar 

  • Goldreich, P., 1986, Final spin states of planets and satellites, Astron. J. 71, 1–7

    Article  Google Scholar 

  • Goldreich, P. and Peale, S., 1966, Spin-orbit coupling in the Solar System, Astron. J. 71, 425–437

    Article  Google Scholar 

  • Goldreich, P. and Toomre, A., 1969, Some remarks on polar wandering, J. of Geophys. Res. 74, 2555–2567

    Article  Google Scholar 

  • Gonczi, R., Froeschlé, Ch. and Froeschlé, CL, 1982, Poynting-Robertson drag on orbital resonance, Icarus 51, 633–654

    Article  Google Scholar 

  • Gonczi, R., Froeschlé, Ch. and Froeschlé, CL, 1983, Trapping time of resonant orbits in presence of Poynting-Robertson drag, in “Dynamical Trapping and Evolution in the Solar System” (Markellos V.V. and Koza Y. eds.), Reidel

    Google Scholar 

  • Gradshteyn, I.S. and Ryzhic, I.M., 1965, Tables of Integrals, Series and Products Academic Press

    Google Scholar 

  • Greenberg, R., 1973, Evolution of satellite resonances by tidal dissipation, Astron. J. 78, 338–346

    Article  Google Scholar 

  • Greenberg, R., 1977, Orbit-orbit resonances in the solar system: varieties and similarities, Vistas in Astronomy 21, 209–239

    Article  Google Scholar 

  • Guckenheimer, J. and Holmes Ph., 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag

    MATH  Google Scholar 

  • Hagihara, Y., 1972, Celestial Mechanics, II: Perturbation Theory. MIT Press

    Google Scholar 

  • Hannay, J.H., 1986, Accuracy loss of action invariance in adiabatic change of a one degree of freedom Hamiltonian, J. Phys. A, 19, 1067–1072

    Article  MathSciNet  Google Scholar 

  • Helwig, G., 1955, Z. Naturforsch. 10A, 508

    Google Scholar 

  • Henrard, J., 1970a, On a perturbation theory using Lie transforms, Celestial Mechanics 1, 111–222

    Google Scholar 

  • Henrard, J., 1970b, Perturbation technique in the theory of nonlinear oscillations and in Celestial Mechanics, Boeing Scient. Res. Lab. Techn. Note 051

    Google Scholar 

  • Henrard, J., 1974, Virtual singularities in the Artificial satellite theory, Celestial Mechanics 10, 437–449

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J., 1982a, Capture into resonance: An extension of the use of the adiabatic invariants, Celestial Mechanics 27, 3–22

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J., 1982b, The adiabatic Invariant: Its use in Celestial Mechanics in applications of modern dynamics (V. Szebehely ed.) pp. 153–171, Reidel, Dordrecht

    Google Scholar 

  • Henrard, J., 1983, On Brown’s conjecture, Celestial Mechanics 31, 115–122

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J., 1985, Spin-orbit resonance and the adiabatic invariant, in “Resonances in the motion of Planets” (Ferras-Mello and Sessin eds.), Univ. Saõ Paulo

    Google Scholar 

  • Henrard, J. and Lemaître, A., 1983a, A second fundamental model for resonance, Celestial Mechanics 30, 197–218

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J. and Lemaître, A., 1983b, A mechanism of formation for the Kirkwood Gaps, Icarus 55, 482–494

    Article  Google Scholar 

  • Henrard, J. and Lemaître, A., 1986a, A perturbative treatment of the 2/1 Jovian resonance, Icarus 69, 266–279

    Article  Google Scholar 

  • Henrard, J. and Lemaître, A., 1986b, A perturbation method for problems with two critical arguments, Celestial Mechanics 39, 213–238

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J. and Murigande, Ch., 1987, Colombo’s top, Celestial Mechanics 40, 345–366

    Article  MathSciNet  MATH  Google Scholar 

  • Henrard, J. and Roels, J., 1974, Equivalence for Lie transforms, Celestial Mechanics 10, 497–512

    Article  MathSciNet  MATH  Google Scholar 

  • Hori, G.I., 1966, Theory of general perturbation with unspecified canonical variables, Pub. Astron. Soc. of Japan 18, 287–296

    Google Scholar 

  • Jeans, J.H., 1924, Cosmogenic problems associated with a secular decrease of mass., M.N.R.A.S. 85, 2

    Google Scholar 

  • Jeans, J.H., 1924, The effect of varying Mass on a binary system, M.N.R.A.S. 85, 912

    Google Scholar 

  • Jefferys, W.H., 1968, Perturbation theory for strongly perturbed dynamical systems, Astron. J. 73, 522–527

    Article  MathSciNet  Google Scholar 

  • Kirchgraber, U. and Stiefel, E., 1978, Methoden der analytischen Störungsrechnung und ihre Anwendungen, B.G. Teubner, Stuttgart

    MATH  Google Scholar 

  • Kneser, H., 1924, Die adiabatisshe Invarianz des Phasenintegrals bei einem Freiheitsgrad, Math. Ann. 91, 156–160

    Google Scholar 

  • Kovrizhnykh, L.M., 1984, Progress in stellarator theory, Plasma Phys. 26, 195–207

    Google Scholar 

  • Krilov, N. and Bogoliubov, N., 1947, Introduction to non-linear mechanics, Annals of Mathematics Studies, II, Princeton U.P.

    Google Scholar 

  • Kruskal, M., 1952, U.S. Atomic Energy Commission Report N40–998 (PM-S-5)

    Google Scholar 

  • Kruskal, M., 1957, Rendiconti del Tezzo Congresso Internazionale sui Fenomeni d’Ion-izzazione nei Gas tenuto a Venezia, Societa Italiana di Fisica, Milan

    Google Scholar 

  • Kruskal, M., 1962, Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic, J. of Math. Phys. 3, 806–828

    Article  MathSciNet  MATH  Google Scholar 

  • Kulsrud, R., 1957, Adiabatic Invariant of the Harmonic Oscillator, Phys. Rev. 106, 205–207

    Article  MATH  Google Scholar 

  • Langevin, P. and De Broglie, M., 1912, La théorie du rayonnement et les quanta, Report on the meeting at the Institute Solvay at Brussels, Gauthier Villars, Paris (p. 450)

    Google Scholar 

  • Lemaître, A., 1984, High-order resonances in the restricted three body problem, Celestial Mechanics 32, 109–126

    Article  MathSciNet  MATH  Google Scholar 

  • Lemaître, A., 1984a, Analysis of a simple mechanism to deplete the Kirkwood Gaps, in Dynamical Trapping and Evolution in the Solar system (V.V. Markellos and Y. Kozai eds) Reidel

    Google Scholar 

  • Lemaître, A., 1984b, L’origine des lacunes de Kirkwood. Thesis, University of Namur

    Google Scholar 

  • Lemaître, A., 1985, The formation of the Kirkwood gaps in the Asteroid Belt, Celestial Mechanics 34, 329–341

    Article  Google Scholar 

  • Lemaître, A. and Henrard, J., 1988, Chaotic Motions in the 2/1 Resonance, in preparation

    Google Scholar 

  • Lenard, A., 1959, Adiabatic Invariance to all orders, Annals of Physics 6, 261–276

    Article  MathSciNet  MATH  Google Scholar 

  • Leung, A. and Meyer, K., 1975, Adiabatic Invariants for Linear Hamiltonian Systems, J. of Dif. Equ. 17, 33–43

    MathSciNet  Google Scholar 

  • Lichtenberg, A.J. and Lieberman, M.A., 1983, Regular and Stochastic Motion, Springer-Verlag

    MATH  Google Scholar 

  • Littlejohn, R.G., 1979, A guiding center Hamiltonian: A new approach, J. Math. Phys. 20, 2445–2458

    Article  MathSciNet  MATH  Google Scholar 

  • Littlejohn, R.G., 1983, Variational principles of guiding center motion, J. Plasma Physics 29, 111–125

    Article  Google Scholar 

  • Littlewood, J.E., 1963, Lorentz’s pendulum problem, Ann. Physics 21, 233–242

    Article  MathSciNet  MATH  Google Scholar 

  • Menyuk, C.R., 1985, Particle motion in the field of a modulated wave, Phys. Rev. A 31, 3282–3290

    Article  Google Scholar 

  • Message, P.J., 1966, On nearly-commensurable periods in the restricted problem of three bodies, Proceedings I AU, Symposium 25, 197–222

    Google Scholar 

  • Meyer, R.E., 1976, Adiabatic variation - Part V: Nonlinear near-periodic oscillator, ZAMP 27, 181–195

    Article  Google Scholar 

  • Meyer, R.E., 1980, Exponential asymptotics, SIAM Review 22, 213–224

    MATH  Google Scholar 

  • Moser, J., 1958, On a generalization of a theorem of A. Liapounoff, Comm. Pure. Applied. Math. XI, 257–271

    Article  Google Scholar 

  • Nayfeh, A.H., 1973, Perturbations Methods, J. Wiley and Sons

    Google Scholar 

  • Neishtadt, A.I., 1975, Passage through a separatrix in a resonance problem with a slowly-varying parameter, Prikl. Matem. Mekhun 39, 621–632

    MathSciNet  Google Scholar 

  • Neishtadt, A.I., 1986, Change in adiabatic invariant at a separatrix, Sov. J. Plasma Phys. 12, 568–573 (Fiz. Plazung 12, 992–1001)

    Google Scholar 

  • Peggs, S.G. and Talman, R.M., 1986, Nonlinear problems in accelerator physics, Ann. Rev. Nucl. Part. Sci. 36, 287–325

    Article  Google Scholar 

  • Peale, S.J., 1969, Generalized Cassini’s laws, Astron. J. 74, 483–489

    Article  Google Scholar 

  • Peale, S.J., 1974, Possible histories of the obliquity of Mercury, Astron. J. 79, 722–744

    Article  Google Scholar 

  • Peale, S.J., 1976, Orbital resonances in the Solar system. Ann. Rev. Astron. Astrophys. 14, 215–246

    Article  Google Scholar 

  • Peale, S.J., 1986, Dynamical evolution of natural satellites: Some examples and consequences, A chapter of the book Natural Satellites (Burns J.A. and Matthews M.S. eds.), Univ. of Arizona Press

    Google Scholar 

  • Poincaré, H., 1899, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars

    MATH  Google Scholar 

  • Poincaré, H., 1902, Sur les Planètes du type d’Hecube, Bull. Astron. 19, 289–310

    Google Scholar 

  • Poincaré, H., 1911, Leçons sur les hypothèses cosmogoniques, Paris

    MATH  Google Scholar 

  • Rüssmjinn, H., 1964, Uber das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 154, 285–300

    Article  MathSciNet  Google Scholar 

  • Schubart, J., 1966, Special cases of the restricted problem of three bodies, Proceedings IAU Symposium 25, 187–193

    Google Scholar 

  • Sessin, W., 1981, Thesis, University of Saõ Paulo

    Google Scholar 

  • Sessin, W. and Ferraz-Mello, S., 1984, Motion of two planets with period commensurable in the ratio 2/1, Celestial Mechanics 32, 307–332

    Article  MathSciNet  MATH  Google Scholar 

  • Siegel, C.L. and Moser, J.K., 1971, Lectures on Celestial Mechanics, Springer-Verlag

    MATH  Google Scholar 

  • Sinclair, A.T., 1972, On the origin of the commensurabilities amongst the satellites of Saturn, Mon. Not. Roy. Astr. Soc. 160, 169–187

    Google Scholar 

  • Sinclair, A.T., 1974, On the origin of the commensurabilities amongst the satellites of Saturn II, Mon. Not. Roy. Astr. Soc. 166, 165–179

    Google Scholar 

  • Stengle, G., 1977, Asymptotic estimates for the adiabatic invariance of a simple oscillator, SIAM J. Math. Anal. 8, 640–651

    Article  MathSciNet  MATH  Google Scholar 

  • Stuart, J., Weidenschilling, S.J. and Davis, D.R., 1985, Orbital resonances in the solar nebula, implications for planetary accretion, Icarus 62, 16–29

    Article  Google Scholar 

  • Stern, D.R, 1971, Classical adiabatic perturbation theory, J. of Math. Phys. 12, 2231–2242

    Article  MATH  Google Scholar 

  • Stoker, J.J., 1950, Nonlinear vibrations, Interscience, New York

    MATH  Google Scholar 

  • Tennyson, J., 1979, In nonlinear dynamics and the beam-beam interaction, A.I.P. Conference Proceedings 57, 158

    Google Scholar 

  • Tennyson, J.L., Cary, J.R. and Escande, D.F., 1986, Change of the adiabatic invariant due to separatrix crossing, Phys. Rev. Letters 56, 2117–2120

    Article  MathSciNet  Google Scholar 

  • Timofeev, A.V., 1978, On the constancy of the adiabatic invariant when the nature of the motion changes, Sov. Phys. JETP 48, 656–659

    Google Scholar 

  • Torbett, M. and Smoluchowski, R., 1980, Sweeping of the Jovian resonances and the evolution of the asteroids, Icarus 44, 722–729

    Article  Google Scholar 

  • Torbett, M. and Smoluchowski, R., 1982, Motion of the Jovian Commensurability resonances and the character of the celestial mechanics in the asteroids zone: Implications for kinematics and structure, Astron. Astrophys. 110, 43–49

    Google Scholar 

  • Urabe, M., 1954, Infinitesimal deformation of the periodic solution of the second kind and its application to the equation of a pendulum, J. Sci. Hiroshima Univ. A18, 183

    MathSciNet  Google Scholar 

  • Urabe, M., 1955, The least upper bound of a damping coefficient ensuring the existence of a periodic motion of a pendulum under constant torque, J. Sci. Hiroshima Univ. A18, 379

    MathSciNet  Google Scholar 

  • Ward, W.R., 1975, Tidal friction and generalized Cassini’s laws in the Solar System, Astron. J. 80, 64–70

    Article  Google Scholar 

  • Ward, W.R., Burns, J.A. and Toon, O.B., 1979, Past obliquity oscillations of Mars: the role of the Tharsis Uplift, J. of Geophys. Res. 84, 243–258

    Article  Google Scholar 

  • Wasow, W., 1976, Adiabatic invariants and the asymptotic theory of ordinary linear differential equations, SIAM-AMS Proceedings 10, 131–144

    MathSciNet  Google Scholar 

  • Wasow, W., 1965, Asymptotic Expansions for Ordinary Differential Equations, J. Wileyand Son, 1965 (p. 128)

    Google Scholar 

  • Wasow, W., 1973, Adiabatic invariance of a simple oscillator, SIAM J. Math. Anal. 4, 78–88

    Article  MathSciNet  MATH  Google Scholar 

  • Weidenschilling, S.J., 1977, The distribution of mass in the planetary system and solar nebula, Astrophys. Space Sci. 51, 153–158

    Article  Google Scholar 

  • Whittaker, E.T., 1916, Proc. Roy. Soc. Edinburgh 37, 95

    Google Scholar 

  • Whittaker, E.T., 1927, Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 3rd. Edition

    MATH  Google Scholar 

  • Wintner, A., 1941, The Analytical Foundations of Celestial Mechanics, Princeton U.P. (p. 24)

    Google Scholar 

  • Wisdom, J., 1983, Chaotic behaviour and the origin of the 3/1 Kirkwood gap, Icarus 56, 51–74

    Article  Google Scholar 

  • Wisdom, J., 1985, A perturbative treatment of motion near the 3/1 Jovian commensurability, Icarus 63, 272–289

    Article  Google Scholar 

  • Yoder, C.F., 1973, On the establishment and evolution of orbit-orbit resonances, Thesis, University of California, Santa-Barbara

    Google Scholar 

  • Yoder, C.F., 1979a, Diagrammatic theory of transition of Pendulum like systems, Celestial Mechanics 19, 3–29

    Article  MathSciNet  MATH  Google Scholar 

  • Yoder, C.F., 1979b, How tidal heating in Io drives the Galilean orbital resonance locks, Nature 279, 747–750

    Article  Google Scholar 

  • Yoder, C.F. and Peale, S.J., 1981, The tides of Io, Icarus 47, 1–35

    Article  Google Scholar 

  • Zeipel, H. von, 1915, Recherche sur le mouvement des petites planètes, Arkiv Math. Astro. Fys. 11, 1–58

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Henrard, J. (1993). The Adiabatic Invariant in Classical Mechanics. In: Jones, C.K.R.T., Kirchgraber, U., Walther, H.O. (eds) Dynamics Reported. Dynamics Reported, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61232-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61232-9_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64755-0

  • Online ISBN: 978-3-642-61232-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics