Recent progress in classical nonlinear dynamics

A. N. Kolmogorov:Dokl. Akad. Nauk SSSR,98, 527 (1954);V. I. Arnol’d:Sov. Math. Dokl.,2, 247 (1961);3, 136 (1962);J. Moser:Nachr. Akad. Wiss. Göttingen: Math. Phys. Kl. IIa,1, 1 (1962); 87 (1955). The following are reviews on the subject:A. N. Kolmogorov:Proceedings of the 1954 International Congress of Mathematics (Amsterdam, 1957), p. 315; English translation inR. Abraham:Foundations of Mechanics, Appendix D (New York, 1967);V. I. Arnol’d:Russian Math. Surveys,18, 9, 85 (1963);J. Moser:Lectures on Hamiltonian systems, inMemoirs of the American Mathematical Society, No. 81 (Providence, R. I., 1968);V. I. Arnol’d andA. Avez:Problèmes ergodiques de la mécanique classique (Paris, 1967);S. Sternberg:Celestial Mechanics, Chap. 3 (New York, 1969);M. Froeschlé:Annales scientifiques de l’Université de Besançon, Mécanique et physique theorique, III serie, fasc. 8, 3 (1968);R. Barrar:Amer. Journ. Math.,98, 206 (1966);Celestial Mechanics,2, 454 (1970);E. M. McMillan: inTopics in Modern Physics, A tribute toE. U. Condon, edited byW. E. Brittin andH. Odabasi (Boulder, Colo., 1971). Ya. G. Sinai:Sov. Math. Dokl.,4, 1818 (1963); inStatistical Mechanics Foundations and Applications, edited byT. A. Bak (New York, 1967), p. 559;Russian Math. Surveys,25, 137 (1970). For an introduction to both KAM theorem and Sinai’s theorem see:A. S. Wightman:Statistical Mechanics at the Turn of the Decade, edited byE. G. D. Cohen (New York, 1971), p. 1;J. L. Lebowitz: inProceedings of the IUPAP Conference on Statistical Mechanics, Chicago, 1971, to be published;J. Ford: inLectures in Statistical Physics, Vol.2, edited byW. C. Schieve (New York, 1972);J. Ford: submitted toAdv. Chem. Phys. In the article quoted in ref. [2],A. S. Wightman says: « The folklore says that the extension of Sinai’s result to a large class of purely repulsive forces will not be difficult but the inclusion of an attraction will very likely result in a breakdown of ergodicity, at least for some energy surfaces. » V. M. Alexeyev:Actes, Congrès International de Mathematiques, Tome 2 (1970), p. 893. H. Poincaré:Méthodes nouvelles de la mécanique céleste (Paris, 1892);E. Fermi:Zeits. Phys.,24, 261 (1923). M. Hénon andC. Heiles:Astron. Journ.,69, 73 (1964); see alsoB. Barbanis:Astron. Journ.,71, 415 (1966). F. G. Gustavson:Astron. Journ.,71, 670 (1966). J. Roels andM. Hénon:Bull. Astron.,2, 267 (1967). G. Contopoulos:Astron. Journ.,68, 3 (1963). L. J. Laslett andK. L. Symon:Comptes Rendu Symposium CERN (Genève, 1956), p. 287. D. E. Ochozinski, V. A. Sarychev, V. A. Zlatonstov andA. P. Torjevski:Kosmicheskie Issledovania,2, 657 (1964). M. Hénon:Compt. Rend.,262, 312 (1966); see alsoI. M. Gel’fand, M. I. Graev, H. M. Zueva, M. S. Michailova andA. I. Morosov:Dokl. Akad. Nauk,143, 81 (1962). V. I. Arnol’d:Russian Math. Surveys,18, 85 (1963). P. A. M. Dirac:The Principles of Quantum Mechanics (Oxford, 1935). For a modern discussion on this point see for exampleP. Bocchieri andA. Loinger:Lett. Nuovo Cimento,4, 310 (1970);1, 709 (1971);2, 41 (1971);P. Bocchieri, A. Crotti andA. Loinger:Lett. Nuovo Cimento, in press. J. Moser andM. Hénon: private communications. E. Fermi, J. Pasta andS. Ulam: Los Alamos Scientific Laboratory Report LA-1940 (1955); reproduced inE. Fermi:Collected Papers (Chicago, 1965), p. 978. P. Bocchieri, A. Scotti, B. Bearzi andA. Loinger:Phys. Rev. A,2, 2013 (1970). F. M. Izrailev, A. I. Khisamutdinov andB. V. Chirikov: preprint 252, Institute of Nuclear Physics, Novosibirsk (1968); translated as Los Alamos Scientific Laboratory Report LA-4440 TR (1970). C. Cercignani, L. Galgani andA. Scotti:Phys. Lett.,38 A, 403 (1972). L. Galgani andA. Scotti:Phys. Rev. Lett.,28, 1173 (1972). W. Nernst:Ver. D. Phys. Ges.,4, 83 (1916). For a short introduction see for exampleY. A. Mitropol’skiy: inNonlinear Differential Equations and Nonlinear Mechanics, edited byJ. P. La Salle andS. Lefschetz (New York, London, 1963), p. 1. See also,S. P. Diliberto:Perturbation Theory of Periodic Surfaces, Mimeographed O.N.R. reports (Berkeley, Cal., 1956–1957). The concept of isolating integral has been introduced byA. Wintner:The Analytical Foundations of Celestial Mechanics (Princeton, N. J., 1947), p. 96. The discussion made here is inspired by a paper ofG. Contopoulos:Astrophys. Journ.,138, 1297 (1963);Astron. Journ.,68, 3 (1963). For another definition, seeD. Lynden-Bell:Mon. Not. Roy. Astron. Soc.,124, 5 (1962). See alsoE. Onofri andM. Pauri:Journ. Math. Phys., in press. For smooth,i.e. C (∞), integrals, one certainly has isolating surfaces for a dense set of values ofc, as guaranteed by Sard’s theorem (see for example,R. Abraham:Foundations of Mechanics (New York, 1967), p. 29). See for exampleE. Hopf:Ergoden Theorie (Berlin, 1937);P. R. Halmos:Lectures in Ergodic Theory (New York, 1956);J. C. Oxtoby andS. M. Ulam:Ann. of Math.,42, 874 (1941);K. Jacobs:Neuere Methoden und Ergebnissen der Ergodentheorien (Berlin, 1960);V. I. Arnol’d andA. Avez:Problèmes ergodiques de la mécanique classique (Paris, 1967);I. E. Farquhar:Ergodic Theory in Statistical Mechanics (London, 1964). See for exampleL. Cesari:Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations, Sect.6.8 (Berlin, 1959). It is recalled here that a function defined in a measurable space and with values in a topological space is called measurable if the inverse image of every open set is a measurable set. See for exampleW. Rudin:Real and Complex Analysis (London, 1970), p. 8. SeeD. Lynden-Bell:Mon. Not. Roy. Astron. Soc.,124, 1 (1962). A classical example of such a situation is the generalization of Maxwell distribution to allow for conservation of angular momentum. A nonclassical example is the distribution of stellar velocities in an axisymmetric stellar system in a state of dynamical equilibrium; for a short review see for exampleA. Ollongren:Ann. Rev. Astron. Astrophys.,3, 113 (1965). D. Lynden-Bell:Mon. Not. Roy. Astr. Soc.,136, 101 (1967); the connection between equilibrium distribution and possibly nonmeasurable integrals is however not very explicit in this paper. B. B. Kadomtsev andO. P. Pogutse:Phys. Rev. Lett.,25, 1155 (1970). The authors thankB. Coppi for having brought this paper, as well as the one quoted in [30], to their attention; an interesting conversation on the subject is also gratefully acknowledged. This concept goes back essentially toJ. W. Gibbs:Elementary Principles in Statistical Mechanics (New York, 1960). See alsoE. Hopf:Journ. Math. Phys.,13, 51 (1934);N. S. Krylov:Works on the Foundations of Statistical Physics (Moscow, 1950) (in Russian). This is not ensured by ergodicity alone. The simplest example of an ergodic nonmixing dynamical (non-Hamiltonian) system is probably the so-called translation of the torus. SeeV. I. Arnol’d andA. Avez:Problèmes ergodiques de la mécanique classique (Paris, 1967), p. 7. Moreover, from this point of view, the property of Poincaré recurrence (L. Cesari:Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations (Berlin, 1959), p. 104) does not constitute a paradox. R. Courant andD. Hilbert:Methods of Mathematical Physics, Vol.1 (New York, 1953), p. 37. E. T. Whittaker:Proc. Roy. Soc. Edinburgh,37, 95 (1916);A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (New York, 1944);J. M. Cherry:Proc. Cambridge Phil. Soc.,22, 325, 510 (1924);Proc. London Math. Soc.,27, 151 (1926);H. Von Zeipel:Ark. Astron. Mat. Fys.,11, No. 1 (1916);G. Contopoulos:Zeits. Astrophys.,49, 273 (1960);A. Deprit:Celestial Mechanics,1, 12 (1969). G. D. Birkhoff:Dynamical Systems (New York, 1927). Strangely enough physicists did not consider the potentialities of this method. Applications to field theory can be found inR. A. Carhart:Journ. Math. Phys.,12, 1748 (1971). The most natural field of application seems to be phonons in solids. SeeM. Kummer:Nuovo Cimento,1 B, 123 (1971);J. Rae:Lett. Nuovo Cimento,3, 520 (1972). C. L. Siegel:Ann. Math.,42, 806 (1941);Vorlesungen über Himmelsmechanik (Berlin, 1956);C. L. Siegel andJ. K. Moser:Lectures on Celestial Mechanics (Berlin, 1971). J. Moser:Comm. Pure Appl. Math.,8, 409 (1955). SeeS. Sternberg:Celestial Mechanics (New York, 1969), p. 129. A more general statement of the theorem is the following. « LetH=H 0(p) +H 1(q, p) be analytic in |Imq i | ⩽h, and |p i | ⩽k, where (1/d)<|∂2 H 0/∂p i ∂p i |<d. Then there is a δ0=δ0(k, h, d) such that if |H 1|<δ<δ0 then \(\mathcal{F} = A_\delta \cup B_\delta \), whereA δ is a union of tori invariant under the flow given byH. The flow restricted to each such torus is (analytically) equivalent to a linear flow. The measure ofB δ tends to zero with δ0. The Hamiltonian of a system of perturbed harmonic oscillators can be brought in the form given in the text by two successive Birkhoff transformations if there are no (coupled) third- and fourth-order resonances. M. Hénon:Bull. Astronom., Ser.3, Tome 1, Fasc. 2, 49 (1966), p. 64. SeeV. M. Alexeyev:Actes, Congrès International de Mathematiques, Tome 2, 893 (1970). See p. 8 of the report quoted in ref. [17]: « … the systems certainly do not show mixing ». I. Newton:Principia (1686), Book 2. This feature has been clearly confirmed byJ. Tuck: unpublished numerical computations at the Los Alamos Scientific Laboratory (1961). G. H. Lunsford andJ. Ford: submitted toJourn. Math. Phys. The present authors take this opportunity to express their gratitude toJ. Ford for having brought to their attention the relevant Russian literature and having kept them informed of results of his research. A work along these lines is in progress. See alsoK. C. Mo:Physica,57, 445 (1972). For a review see for exampleM. D. Kruskal: inProceedings of the IBM Scientific Computing Symposium on Large-Scale Problems in Physics, IBM Data Processing Division (New York, 1965), p. 43. C. S. Gardner, J. M. Greene, M. D. Kruskal andR. M. Miura:Phys. Rev. Lett.,19, 1095 (1967);R. M. Miura, C. S. Gardner andM. D. Kruskal:Journ. Math. Phys.,9, 1204 (1968). In this connection, see alsoJ. Toda:Journ. Phys. Soc. Japan,23, 501 (1967). In this connection, the example considered byD. Lynden-Bell [30] should be of interest. See ref. [21]. For example, for A, Kr, Xe, N2 and O2 one hasm 1/2 ɛ 1/2 σ/Zħ = 2.02, 1.86, 1.97, 2.03 and 1.94 respectively. The valuesE c i should be related to the radii of convergence of the Birkhoff transformation. See ref. [8]. See ref. [20], where the expression for the entropy should be divided byJ c, under the hypothesisJ c i =J c. One expectsE c i =αω i , whereα is a characteristic action of the system. Considerations supporting this on the basis of the numerical results of ref. [6] will be published shortly.