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Mathematics in Engineering

2021, Volume 3, Issue 6: 1-24. doi: 10.3934/mine.2021045
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Foundations of physics in Milan, Padua and Paris. Newtonian trajectories from celestial mechanics to atomic physics

  • Department of Mathematics "F. Enriques", University of Milan, via Saldini 50, 20133 Milan, Italy
  • Received: 06 March 2020 Accepted: 28 August 2020 Published: 09 November 2020

  • This paper is written, in a very informal and colloquial style, on the occasion of the seventieth birthday of Antonio Giorgilli. The aim is to describe how his first scientific works were actually conceived within a group that happened to be formed in the years seventies with an ambitious program on the foundations of physics. Namely, to understand whether the recent (at those times) progress in dynamical systems theory might allow one to enlighten in some new way the relations between quantum mechanics and classical physics. This required to understand what impact dynamical systems theory may have on the foundations of classical statistical mechanics (with particular attention to the Fermi-Pasta-Ulam problem), and on matter-radiation interaction. In such a frame Celestial Mechanics too started to be addressed, particularly by Antonio, initially just as a kind of a byproduct. Here a recollection is given of how the group was formed. Then a quick review is given of the results obtained, the attention being mainly addressed to those relevant for the original foundational program.

    Citation: L. Galgani. Foundations of physics in Milan, Padua and Paris. Newtonian trajectories from celestial mechanics to atomic physics[J]. Mathematics in Engineering, 2021, 3(6): 1-24. doi: 10.3934/mine.2021045

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  • This paper is written, in a very informal and colloquial style, on the occasion of the seventieth birthday of Antonio Giorgilli. The aim is to describe how his first scientific works were actually conceived within a group that happened to be formed in the years seventies with an ambitious program on the foundations of physics. Namely, to understand whether the recent (at those times) progress in dynamical systems theory might allow one to enlighten in some new way the relations between quantum mechanics and classical physics. This required to understand what impact dynamical systems theory may have on the foundations of classical statistical mechanics (with particular attention to the Fermi-Pasta-Ulam problem), and on matter-radiation interaction. In such a frame Celestial Mechanics too started to be addressed, particularly by Antonio, initially just as a kind of a byproduct. Here a recollection is given of how the group was formed. Then a quick review is given of the results obtained, the attention being mainly addressed to those relevant for the original foundational program.


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    [1] D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilinesr PDEs, Ann. Scuola Norm. Sup. Pisa Cl. Sci., IV (2005), 669-702.
    [2] D. Bambusi, B. Langella, A C∞ Nekhoroshev theorem, Mathematics in Engineering, 3 (2020), 1-17.
    [3] D. Bambusi, A. Ponno, On metastability in FPU, Commun. Math. Phys., 264 (2006), 539-561.
    [4] G. Benettin, The elements of Hamiltonian perturbation theory, In: D. Benest, C. Froesché, E. Lega, Hamiltonian systems and frequency analysis, Cambridge Sci. Pub., 2004.
    [5] G. Benettin, A. Carati, L. Galgani, A. Giorgilli, The Fermi-Pasta-Ulam problem and the metastability perspective, Berlin: Springer, 2007.
    [6] G. Benettin, H. Christodoulidi, A. Ponno, The Fermi-Pasta-Ulam problem and its underlying integrable dynamics, J. Stat. Phys., 152 (2013), 195-212.
    [7] G. Benettin, F. Fassò, Fast rotations of the symmetric rigid body: A general study by Hamiltonian perturbation theory. Part I, Nonlinearity, 9 (1996), 137-186.
    [8] G. Benettin, F. Fassò, M. Guzzo, Fast rotations of the symmetric rigid body: A study by Hamiltonian perturbation theory. Part Ⅱ, Gyroscopic rotations, Nonlinearity, 10 (1997), 1695-1717.
    [9] G. Benettin, F. Fassò, M. Guzzo, Nekhoroshev-stability of L4 and L5 in the spatial restricted three-body problem, Regul. Chaotic Dyn., 3 (1998), 56-72.
    [10] G. Benettin, F. Fassò, M. Guzzo, Long term stability of proper rotations of the perturbed Euler rigid body, Commun. Math. Phys., 250 (2004), 133-160.
    [11] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory — Part 2: Numerical application, Meccanica, 15 (1980), 9-30.
    [12] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method, Il Nuovo Cimento B, 79 (1984), 201-223.
    [13] G. Benettin, L. Galgani, A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Cel. Mech., 37 (1985), 1-25.
    [14] G. Benettin, L. Galgani, J. M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys. Rev. A, 14 (1976), 2338-2345.
    [15] G. Benettin, M. Guzzo, V. Marini, Adiabatic chaos in the spin orbit problem, Celest. Mech. Dyn. Astr., 101 (2008), 203.
    [16] G. Benettin, A. Ponno, Time-scales to equipartition in the Fermi-Pasta-Ulam problem: Finite-size effects and thermodynamic limit, J. Stat. Phys., 144 (2011), 793.
    [17] G. Benettin, A. Ponno, Understanding the FPU state in FPU-like models, Mathematics in Engineering, 3 (2020), 1-22.
    [18] L. Berchialla, L. Galgani, A. Giorgilli, Localization of energy in FPU chains, Discrete Cont. Dyn. A, 11 (2004), 855-866.
    [19] P. Bocchieri, A. Scotti, B. Bearzi, A. Loinger, Anharmonic chain with Lennard-Jones interaction, Phys. Rev. A, 1970, 2013-2019.
    [20] A. Carati, Pair production in classical electrodynamics, Found. Phys., 28 (1998), 843-853.
    [21] A. Carati, Thermodynamics and time averages, Physica A, 348 (2005), 110-120.
    [22] A. Carati, On the definition of temperature using time-averages, Physica A, 369 (2006), 417-431.
    [23] A. Carati, An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit, J. Stat. Phys., 128 (2007), 1057-1077.
    [24] A. Carati, F. Benfenati, A. Maiocchi, M. Zuin, L. Galgani, Chaoticity threshold in magnetized plasmas: Numerical results in the weak coupling regime, Chaos, 24 (2014), 013118.
    [25] A. Carati, S. Cacciatori, L. Galgani, Discrete matter, far fields and dark matter, EPL, 83 (2008), 59002.
    [26] A. Carati, S. Cacciatori, L. Galgani, Far fields, from electrodynamics to gravitation, and the dark matter problem, In: Chaos in Astronomy. Astrophysics and Space Science Proceedings, Berlin, Heidelberg: Springer, 2008,325-335.
    [27] A. Carati, L. Galgani, Nonradiating normal modes in a classical many-body model of matterradiation interaction, Il Nuovo Cimento B, 8 (2003), 839-851.
    [28] A. Carati, L. Galgani, Far fields as a possible substitute for dark matter, In: Chaos, diffusion and nonintegrability in Hamiltonian systems, La Plata, 2012.
    [29] A. Carati, L. Galgani, Classical microscopic theory of dispersion, emission and absorption of light in dielectrics, Eur. Phys. J. D, 68 (2014), 307.
    [30] A. Carati, L. Galgani, Progress along the lines of the Einstein Classical Program: An enquiry on the necessity of quantization in light of the modern theory of dynamical systems. Available from: http://www.mat.unimi.it/users/galgani.
    [31] A. Carati, L. Galgani, F. Gangemi, R. Gangemi, Relaxation times and ergodic properties in a realistic ionic-crystal model, and the modern form of the FPU problem, Physica A, 532 (2019), 121911.
    [32] A. Carati, L. Galgani, F. Gangemi, R. Gamgemi, Electronic trajectories in atomic physics: The chemical bond in the $H_2.+$ ion, Chaos, 30 (2020), 063109.
    [33] A. Carati, L. Galgani, A. Giorgilli, The Fermi-Pasta-Ulam problem as a challenge for the foundations of physics, Chaos, 15 (2005), 015105.
    [34] A. Carati, L. Galgani, A. Maiocchi, F. Gangemi, R. Gangemi, The FPU problem as a statisticalmechanical counterpart of the KAM problem, and its relevance for the foundations of physics, Regul. Chaotic Dyn., 23 (2018), 704-719.
    [35] A. Carati, L. Galgani, A. Maiocchi, F. Gangemi, R. Gangemi, Classical infrared spectra of ionic crystals and their relevance for statistical mechanics, Physica A, 506 (2018), 1-10.
    [36] A. Carati, A. Maiocchi, Exponentially long stability times for a nonlinear lattice in the thermodynamic limit, Commun. Math. Phys., 314 (2012), 129-161.
    [37] A. Carati, M. Zuin, A. Maiocchi, M. Marino, E. Martines, L. Galgani, Transition from order to chaos, and density limit, in magnetized plasmas, Chaos, 22 (2012), 033124.
    [38] C. Cercignani, L. Galgani, A. Scotti, Zero-point energy in classical non-linear mechanics, Phys. Lett. A, 38 (1972), 403-404.
    [39] G. Contopoulos, A review of the "Third" integral, Mathematics in Engineering, 2 (2020), 472-511.
    [40] W. De Roeck, F. Huveneers, Asymptotic localization of energy in nondisordered oscillator chains, Commun. Pure Appl. Math., 68 (2015), 1532-1568.
    [41] E. Diana, L. Galgani, A. Giorgilli, A. Scotti, On the direct construction of integrals of Hamiltonian systems near an equilibrium point, Boll. U. M. I., 11 (1975), 84-89.
    [42] T. Erber, Mathematical Reviews MR1652395, 2000a: 78006.
    [43] F. Fassò, M. Guzzo, G. Benettin, Nekhoroshev stability of elliptic equilibria of Hamiltonian systems, Commun. Math. Phys., 197 (1998), 347-360.
    [44] L. Galgani, Carlo Cercignani's interests for the foundations of physics, Meccanica, 47 (2012), 1723-1735.
    [45] L. Galgani, A. Scotti, Planck-like distribution in classical nonlinear mechanics, Phys. Rev. Lett., 28 (1972), 1173-1176.
    [46] L. Galgani, A. Scotti, Recent progress in classical nolinear dynamics, La Rivista del Nuovo Cimento, 2 (1972), 189-209.
    [47] F. Gangemi, A. Carati, L. Galgani, R. Gangemi, A. Maiocchi, Agreement of classical Kubo theory with the infrared dispersion curves n(ω) of ionic crystals, EPL, 110 (2015), 47003.
    [48] C. S. Gardner, J. M. Green, M. D. Kruskal, R. M. Miura, Korteweg-devries equation and generalizations. VI. Methods for exact solutions, Commun. Pure Appl. Math., 27 (1974), 97-133.
    [49] A. Giorgilli, A computer program for integrals of motion, Comput. Phys. Commun., 16 (1979), 331-343.
    [50] A. Giorgilli, Rigorous results on the power expansions for the integrals of a hamiltonian system near an elliptic equilibrium point, Ann. Inst. H. Poincaré, 48 (1988), 423-439.
    [51] A. Giorgilli, Perturbation methods in celestial mechanics, In: Satellite dynamics and space missions, Springer INDAM Series, 2019, 51-114.
    [52] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani, C. Simó, Effective stability for a hamiltonian system near an elliptic equilibrium point, with an applicatiion to the restricted three body problem, J. Differ. Equations, 77 (1989), 167-198.
    [53] A. Giorgilli, L. Galgani, Formal integrals of motions for an autonomous Hamiltonian system near an equilibrium point, Cel. Mech., 17 (1978), 267-280.
    [54] A. Giorgilli, U. Locatelli, Kolmogorov theorem and classical perturbation theory, Z. Angew. Math. Phys., 48 (1997), 220-261.
    [55] A. Giorgilli, U. Locatelli, On classical series expansion for quasi-periodic motions, Math. Phys. Electron. J., 3 (1997), 1-25.
    [56] A. Giorgilli, U. Locatelli, M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Discrete Cont. Dyn. A, 104 (2009), 159-173.
    [57] A. Giorgilli, U. Locatelli, M. Sansottera, On the convergence of an algorithm constructing of the normal form for lower dimesionality elliptic tori in planetary systems, Discrete Cont. Dyn. A, 119 (2014), 397-424.
    [58] A. Giorgilli, S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Cont. Dyn. S, 3 (2010), 601-621.
    [59] A. Giorgilli, S. Paleari, T. Penati, Extensive adiabatic invariants for nonlinear chains, J. Stat. Phys., 148 (2012), 1106-1134.
    [60] A. Giorgilli, S. Paleari, T. Penati, An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit, Ann. I. H. Poincaré PR, 16 (2015), 897-959.
    [61] M. Guzzo, L. Chierchia, G. Benettin, The steep Nekhoroshev's theorem, Commun. Math. Phys., 342 (2016), 569-601.
    [62] M. Guzzo, F. Fassò, G. Benettin, On the stability of elliptic equilibria, Math. Phys. Electron. J., 4 (1998), 1-16.
    [63] M. Guzzo, A. Morbidelli, Construction of a Nekhoroshev-like result for the asteroid belt dynamical system, Discrete Cont. Dyn. A, 66 (1996), 255-292.
    [64] A. Lerose, A. Sanzeni, A. Carati, L. Galgani, Classical microscopic theory of polaritons in ionic crystals, Eur. Phys. J. D, 68 (2014), 35.
    [65] U. Locatelli, A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, Discrete Cont. Dyn. B, 7 (2007), 377-398.
    [66] U. Locatelli, A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Discrete Cont. Dyn. A, 78 (2000), 47-74.
    [67] A. Maiocchi, Freezing of the optical-branch energy in a diatomic FPU chain, Commun. Math. Phys., 372 (2019), 91-117.
    [68] A. Maiocchi, D. Bambusi, A. Carati, An averaging theorem for FPU in the thermodynamic limit, J. Stat. Phys., 155 (2014), 300-322.
    [69] A. Morbidelli, A. Giorgilli, Superexponential stability of KAM tori, J. Stat. Phys., 78 (1995), 1607-1617.
    [70] A. Morbidelli, M. Guzzo, The Nekhoroshev thorem and the asteroid belt dynamical system, Discrete Cont. Dyn. A, 65 (1996), 107-136.
    [71] J. Moser, Mathematical Reviews MR0097508, 20 n. 4066.
    [72] N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1-65.
    [73] A. Ponno, L. Galgani, F. Guerra, Analytical estimate of stochasticity thresholds in Fermi-PastaUlam and φ4 models, Phys. Rev. E, 61 (2000), 7081-7086.
    [74] M. Sansottera, A. Giorgilli, T. Carletti, High-order control for symplectic maps, Physica D, 316 (2016), 1-15.
    [75] P. A. Schilpp, Albert Einstein, Philosopher-scientist, Library of Living Philosophers, Volume VⅡ, Northwestern University, 1949.
    [76] M. Volpi, U. Locatelli, M. Sansottera, A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems, Discrete Cont. Dyn. A, 130 (2018), 36.
    [77] J. A. Wheeler, R. P. Feynman, Interaction with the absorber as the mechanism of radiation, Rev. Mod. Phys., 17 (1945), 157-181.
    [78] J. A. Wheeler, R. P. Feynman, Classical electrodynamics in terms of direct interparticle action, Rev. Mod. Phys., 21 (1949), 425-433.
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