Terracini: Le traiettorie paraboliche della meccanica celeste come transizioni di fase minimali

Terracini, Susanna:
Le traiettorie paraboliche della meccanica celeste come transizioni di fase minimali
Bollettino dell'Unione Matematica Italiana Serie 9 5 (2012), fasc. n.3, p. 689-710, (Italian)
pdf (647 Kb), djvu (270 Kb). | MR 3051740 | Zbl 1282.70029

Sunto

Quanto segue è il testo della conferenza plenaria che ho tenuto al XVIII Congresso dell'Unione Matematica Italiana, in cui ho esposto il contenuto di due lavori in collaborazione con V. Barutello e G. Verzini ([2, 3]). In tali lavori si è sviluppato l'approccio variazionale alle traiettorie paraboliche della Meccanica Celeste, che connettono due configurazioni centrali minimali.

Referenze Bibliografiche

[1] V. BARUTELLO - D. L. FERRARIO - S. TERRACINI, On the singularities of generalized solutions to n-body-type problems, Int. Math. Res. Not. IMRN (2008). | fulltext (doi) | MR 2439573 | Zbl 1143.70005

[2] V. BARUTELLO - S. TERRACINI - G. VERZINI, Entire Minimal Parabolic Trajectories: the planar anisotropic Kepler problem, Arch. Ration. Mech. Anal., to appear (2011). | fulltext (doi) | MR 3005324 | Zbl 1320.70006

[3] V. BARUTELLO - S. TERRACINI - G. VERZINI, Entire Parabolic Trajectories as Minimal Phase Transitions, preprint (2011). | fulltext (doi) | MR 3148122 | Zbl 1287.70007

[4] V. BENCI, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 401-412. | fulltext EuDML | MR 779876 | Zbl 0588.35007

[5] J. CHAZY, Sur certaines trajectoires du problème des n corps, Bulletin Astronomique, 35 (1918), 321-389.

[6] J. CHAZY, Sur l'allure du mouvement dans le problème de trois corps quand le temps crois indèfinimment, Ann. Sci. Ec. Norm. Sup., 39 (1922), 29-130. | fulltext EuDML | MR 1509241 | Zbl 48.1074.04

[7] K.-C. CHEN, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 158 (2001), 293-318. | fulltext (doi) | MR 1847429 | Zbl 1028.70009

[8] K.-C. CHEN, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Ann. of Math. (2), 167 (2008), 325-348. | fulltext (doi) | MR 2415377 | Zbl 1170.70006

[9] K.-C. CHEN, Variational constructions for some satellite orbits in periodic gravitational force fields, Amer. J. Math., 132 (2010), 681-709. | fulltext (doi) | MR 2666904 | Zbl 1250.70012

[10] A. CHENCINER, Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des n corps, Regul. Chaotic Dyn., 3 (1998), 93-106. | fulltext (doi) | MR 1704972 | Zbl 0973.70011

[11] A. CHENCINER - R. MONTGOMERY, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. (2), 152 (2000). | fulltext EuDML | fulltext (doi) | MR 1815704 | Zbl 0987.70009

[12] L. CHIERCHIA - G. PINZARI, The planetary N-body problem: Symplectic foliation, reductions and invariant tori, Invent. Math. (2011). | fulltext (doi) | MR 2836051 | Zbl 1316.70010

[13] F. H. CLARKE - R. B. VINTER, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98. | fulltext (doi) | MR 779053 | Zbl 0563.49009

[14] A. DA LUZ - E. MADERNA, On the free time minimizers of the newtonian n-body problem, Math. Proc. Cambridge Philos. Soc., to appear (2011). | fulltext (doi) | MR 3177865 | Zbl 1331.70035

[15] R. L. DEVANEY, Collision orbits in the anisotropic Kepler problem, Invent. Math., 45 (1978), 221-251. | fulltext EuDML | fulltext (doi) | MR 495360 | Zbl 0382.58015

[16] R. L. DEVANEY, Singularities in classical mechanical systems, in Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), vol. 10 of Progr. Math., Birkhäuser Boston, Mass., 1981, 211-333. | MR 633766

[17] A. FATHI, Weak Kam Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007.

[18] A. FATHI - E. MADERNA, Weak KAM theorem on non compact manifolds, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 1-27. | fulltext (doi) | MR 2346451

[19] A. FATHI - A. SICONOLFI, Existence of $C^1$C1 critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. | fulltext (doi) | MR 2031431 | Zbl 1061.58008

[20] D. L. FERRARIO, Symmetry groups and non-planar collisionless action-minimizing solutions of the three-body problem in three-dimensional space, Arch. Ration. Mech. Anal., 179 (2006), 389-412. | fulltext (doi) | MR 2208321 | Zbl 1138.70322

[21] D. L. FERRARIO, Transitive decomposition of symmetry groups for the n-body problem, Adv. Math., 213 (2007), 763-784. | fulltext (doi) | MR 2332609 | Zbl 1114.70013

[22] D. L. FERRARIO - A. PORTALURI, On the dihedral n-body problem, Nonlinearity, 21 (2008), 1307-1321. | fulltext (doi) | MR 2422381 | Zbl 1138.70007

[23] D. L. FERRARIO - S. TERRACINI, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. | fulltext (doi) | MR 2031430 | Zbl 1068.70013

[24] G. FUSCO - G. F. GRONCHI - P. NEGRINI, Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem, Invent. Math., 185 (2011), 283-332. | fulltext (doi) | MR 2819162 | Zbl 1305.70023

[25] W. B. GORDON, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975), 113-135. | fulltext (doi) | MR 377983 | Zbl 0276.58005

[26] W. B. GORDON, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. | fulltext (doi) | MR 502484 | Zbl 0378.58006

[27] M. C. GUTZWILLER, The anisotropic Kepler problem in two dimensions, J. Mathematical Phys., 14 (1973), 139-152. | fulltext (doi) | MR 349203

[28] M. C. GUTZWILLER, Bernoulli sequences and trajectories in the anisotropic Kepler problem, J. Mathematical Phys., 18 (1977), 806-823. | fulltext (doi) | MR 459107

[29] J. K. HALE - H. KOÇAK, Dynamics and bifurcations, vol. 3 of Texts in Applied Mathematics, Springer-Verlag, New York, 1991. | fulltext (doi) | MR 1138981

[30] N. D. HULKOWER - D. G. SAARI, On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem, J. Differential Equations, 41 (1981), 27-43. | fulltext (doi) | MR 626619 | Zbl 0475.70010

[31] M. KLEIN - A. KNAUF, Classical planar scattering by coulombic potentials, Lecture Notes in Physics Monographs, Springer-Verlag, Berlin, 1992. | MR 3752660 | Zbl 0783.70001

[32] A. KNAUF, The n-centre problem of celestial mechanics for large energies, J. Eur. Math. Soc. (JEMS), 4 (2002), 1-114. | fulltext EuDML | fulltext (doi) | MR 1891507 | Zbl 1010.70011

[33] T. LEVI-CIVITA, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. | fulltext (doi) | MR 1555161 | Zbl 47.0837.01

[34] E. MADERNA - A. VENTURELLI, Globally minimizing parabolic motions in the Newtonian N-body problem, Arch. Ration. Mech. Anal., 194 (2009), 283-313. | fulltext (doi) | MR 2533929 | Zbl 1253.70015

[35] E. MADERNA, On weak kam theory for N-body problems, Ergod. Th. & Dynam. Sys., to appear (2011). | fulltext (doi) | MR 2995654

[36] C. MARCHAL, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom., 83 (2002), 325-353. | fulltext (doi) | MR 1956531 | Zbl 1073.70011

[37] C. MARCHAL - D. G. SAARI, On the final evolution of the n-body problem, J. Differential Equations, 20 (1976), 150-186. | fulltext (doi) | MR 416150 | Zbl 0336.70010

[38] R. MCGEHEE, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. | fulltext EuDML | fulltext (doi) | MR 359459 | Zbl 0297.70011

[39] R. MOECKEL, Chaotic dynamics near triple collision, Arch. Rational Mech. Anal., 107 (1989), 37-69. | fulltext (doi) | MR 1000223 | Zbl 0697.70021

[40] C. MOORE, Braids in Classical Dynamics, Phys. Rev. Lett., 70, no. 24 (1993), 3675-3679. | fulltext (doi) | MR 1220207 | Zbl 1050.37522

[41] J. MOSER, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure. Appl. Math., 23 (1970), 609-636. | fulltext (doi) | MR 269931 | Zbl 0193.53803

[42] H. POLLARD, The behavior of gravitational systems, J. Math. Mech., 17 (1967/1968), 601-611. | fulltext (doi) | MR 261826 | Zbl 0159.26102

[43] H. POLLARD, Celestial mechanics, Mathematical Association of America, Washington, D. C., 1976. | MR 434057 | Zbl 0353.70009

[44] D. G. SAARI, Expanding gravitational systems, Trans. Amer. Math. Soc., 156 (1971), 219-240. | fulltext (doi) | MR 275729 | Zbl 0215.57001

[45] D. G. SAARI, The manifold structure for collision and for hyperbolic-parabolic orbits in the n-body problem, J. Differential Equations, 55 (1984), 300-329. | fulltext (doi) | MR 766126 | Zbl 0571.70009

[46] M. SHIBAYAMA, Multiple symmetric periodic solutions to the 2n-body problem with equal masses, Nonlinearity, 19 (2006), 2441-2453. | fulltext (doi) | MR 2260271 | Zbl 1260.70006

[47] M. SHIBAYAMA, Minimizing periodic orbits with regularizable collisions in the n-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. | fulltext (doi) | MR 2771668 | Zbl 1291.70049

[48] S. TERRACINI - A. VENTURELLI, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Ration. Mech. Anal., 184 (2007), 465-493. | fulltext (doi) | MR 2299759 | Zbl 1111.70010

[49] A. VENTURELLI, Une caractérisation variationelle des solutions de Lagrange du problème plan des trois corps, Comp. Rend. Acad. Sci. Paris, 332, Série I (2001), 641-644. | fulltext (doi) | MR 1841900 | Zbl 1034.70007

[50] E. T. WHITTAKER, A treatise on the analytical dynamics of particles and rigid bodies: With an introduction to the problem of three bodies, 4th ed, Cambridge University Press (New York, 1959), xiv+456. | MR 103613

[51] J. WALDVOGEL, Quaternions for regularizing celestial mechanics: the right way, Celestial Mech. Dynam. Astronom., 102 (2008), 149-162. | fulltext (doi) | MR 2452904 | Zbl 1154.70309

bdim: Biblioteca Digitale Italiana di Matematica

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Referenza completa

Sunto

Referenze Bibliografiche