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Normal form for lower dimensional elliptic tori in Hamiltonian systems

  • Received: 04 March 2021 Accepted: 16 October 2021 Published: 23 November 2021
  • We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.

    Citation: Chiara Caracciolo. Normal form for lower dimensional elliptic tori in Hamiltonian systems[J]. Mathematics in Engineering, 2022, 4(6): 1-40. doi: 10.3934/mine.2022051

    Related Papers:

  • We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.



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    [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russ. Math. Surv., 18 (1963), 85–191.
    [2] L. Berchialla, L. Galgani, A. Giorgilli, Localization of energy in FPU chains, Discrete Cont. Dyn. Syst., 11 (2004), 855–866. doi: 10.3934/dcds.2004.11.855
    [3] M. Berti, L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Commun. Math. Phys., 305 (2011), 741–796. doi: 10.1007/s00220-011-1264-3
    [4] L. Biasco, L. Chierchia, E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91–135. doi: 10.1007/s00205-003-0269-2
    [5] L. Biasco, L. Chierchia, E. Valdinoci, N-dimensional elliptic invariant tori for the planar (N+1)-body problem, SIAM J. Math. Anal., 37 (2006), 1560–1588. doi: 10.1137/S0036141004443646
    [6] C. Caracciolo, U. Locatelli, Computer-assisted estimates for Birkhoff normal form, J. Comput. Dyn., 7 (2020), 425–460. doi: 10.3934/jcd.2020017
    [7] C. Caracciolo, U. Locatelli, Elliptic tori in FPU chains with a small number of nodes, Commun. Nonlinear Sci. Numer. Simul., 97 (2021), 105759. doi: 10.1016/j.cnsns.2021.105759
    [8] C. Caracciolo, U. Locatelli, M. Sansottera, M. Volpi, Librational KAM tori in the secular dynamics of the $\upsilon$–Andromedæ planetary system, arXiv: 2108.11834.
    [9] A. Celletti, L. Chierchia, Rigorous estimates for a Computer-assisted KAM theory, J. Math. Phys., 28 (1987), 2078–2086. doi: 10.1063/1.527418
    [10] A. Celletti, L. Chierchia, KAM stability and celestial mechanics, Volume 187 of {Memoirs of the American Mathematical Society}, 2007,878.
    [11] A. Celletti, A. Giorgilli, U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems, Nonlinearity, 13 (2000), 397–412. doi: 10.1088/0951-7715/13/2/304
    [12] L. Chierchia, C. Falcolini, A direct proof of a theorem by Kolmogorov in Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 21 (1994), 541–593.
    [13] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1988), 115–147.
    [14] L. H. Eliasson, Absolutely convergent series expansion for quasi–periodic motions, MPEJ, 3 (1996), 1–33.
    [15] E. Fermi, J. Pasta, S. Ulam, Studies of Nonlinear Problems, Los Alamos Report, 1955, LA-1940. reprinted in [17].
    [16] G. Gallavotti, Twistless KAM tori, Commun. Math. Phys., 164 (1994), 145–156. doi: 10.1007/BF02108809
    [17] G. Gallavotti, The Fermi-Pasta-Ulam problem: A status report, Berlin: Springer, 2008.
    [18] G. Gentile, V. Mastropietro, Methods of analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications, Rev. Math. Phys., 8 (1996), 393–444. doi: 10.1142/S0129055X96000135
    [19] A. Giorgilli, Quantitative methods in classical perturbation theory, In: Proceedings of the Nato ASI school "From Newton to chaos: modern techniques for understanding and coping with chaos in N–body dynamical systems", New York: Plenum Press, 1995, 21–37.
    [20] A. Giorgilli, Notes on exponential stability of Hamiltonian systems, In: Dynamical systems, Part I, Pubbl. Cent. Ric. Mat. Ennio De Giorgi, Sc. Norm. Sup. Pisa, 2003, 87–198
    [21] A. Giorgilli, U. Locatelli, Kolmogorov theorem and classical perturbation theory, Z. angew. Math. Phys., 48 (1997), 220–261. doi: 10.1007/PL00001475
    [22] A. Giorgilli, U. Locatelli, On classical series expansion for quasi-periodic motions, MPEJ, 3 (1997), 1–25.
    [23] A. Giorgilli, U. Locatelli, A classical self–contained proof of Kolmogorov's theorem on invariant tori, In: Proceedings of the NATO ASI school "Hamiltonian systems with three or more degrees of freedom", Dordrecht–Boston–London: Kluwer Academic Publishers, 1999, 72–89.
    [24] A. Giorgilli, U. Locatelli, M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celest. Mech. Dyn. Astr., 104 (2009), 159–173. doi: 10.1007/s10569-009-9192-7
    [25] A. Giorgilli, U. Locatelli, M. Sansottera, On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems, Celest. Mech. Dyn. Astr., 119 (2014), 397–424. doi: 10.1007/s10569-014-9562-7
    [26] A. Giorgilli, U. Locatelli, M. Sansottera, Improved convergence estimates for the Schröder–Siegel problem, Ann. Mat., 194 (2015), 995–1013. doi: 10.1007/s10231-014-0408-4
    [27] A. Giorgilli, S. Marmi, Convergence radius in the Poincaré–Siegel problem, Discrete Cont. Dyn. Sys. S, 3 (2010), 601–621.
    [28] A. Giorgilli, A. Morbidelli, Invariant KAM tori and global stability for Hamiltonian systems, Z. angew. Math. Phys., 48 (1997), 102–134. doi: 10.1007/PL00001462
    [29] A. Giorgilli, M. Sansottera, Methods of algebraic manipulation in perturbation theory, In: "Chaos, Diffusion and Non-integrability in Hamiltonian Systems – Applications to Astronomy", Proceedings of the Third La Plata International School on Astronomy and Geophysics, La Plata: Universidad Nacional de La Plata and Asociación Argentina de Astronomía Publishers, 2012,102–134
    [30] A. Giorgilli, Ch. Skokos, On the stability of the Trojan asteroids, Astron. Astrophys., 317 (1997), 254–261.
    [31] W. Gröbner, Die Lie-Reihen und Ihre Anwendungen, Berlin: Springer Verlag, 1960.
    [32] U. Locatelli, A. Giorgilli, Invariant tori in the secular motions of the three–body planetary systems, Celest. Mech. Dyn. Astr., 78 (2000), 47–74. doi: 10.1023/A:1011139523256
    [33] U. Locatelli, E. Metetlidou, Convergence of Birkhoff normal form for essentially isochronous systems, Meccanica, 33 (1998), 195–211. doi: 10.1023/A:1004319215392
    [34] A. Luque, J. Villanueva, A KAM theorem without action-angle variables for elliptic lower-dimensional tori, Nonlinearity, 24 (2011), 1033–1080. doi: 10.1088/0951-7715/24/4/003
    [35] V. K. Melnikov, On some cases of conservation of almost periodic motions with a small change of the Hamiltonian function, Dokl. Akad. Nauk SSSR, 165 (1965), 1245–1248.
    [36] A. Morbidelli, A. Giorgilli, Superexponential stability of KAM tori, J. Stat. Phys., 78 (1995), 1607–1617. doi: 10.1007/BF02180145
    [37] J. Moser, Convergent series expansions for quasi-periodic motion, Math. Ann., 169, (1967), 137–176.
    [38] N. N. Nekhoroshev, An exponential estimates of the stability time of near–integrable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1.
    [39] N. N. Nekhoroshev, Exponential estimates of the stability time of near–integrable Hamiltonian systems Ⅱ, (Russian), Trudy Sem. Petrovs., 5 (1979), 5–50.
    [40] J. Pöschel, On elliptic lower dimensional tori in Hamiltonian sytems, Math. Z., 202 (1989), 559–608. doi: 10.1007/BF01221590
    [41] J. Pöschel, A KAM-theorem for some nonlinear PDEs, Ann. Scuola Norm., 23 (1996), 119–148.
    [42] M. Sansottera, V. Danesi, Kolmogorov variation: KAM with knobs (à la Kolmogorov), arXiv: 2109.06345.
    [43] M. Sansottera, U. Locatelli, A. Giorgilli, A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems, Celest. Mech. Dyn. Astr., 111 (2011), 337–361. doi: 10.1007/s10569-011-9375-x
    [44] M. Sansottera, U. Locatelli, A. Giorgilli, On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system, Math. Comput. Simulat., 88 (2013), 1–14. doi: 10.1016/j.matcom.2010.11.018
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