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