Doubly asymptotic trajectories of Lagrangian systems and a problem by Kirchhoff

Maria Letizia Bertotti; Sergey V. Bolotin

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1997)

  • Volume: 8, Issue: 2, page 93-100
  • ISSN: 1120-6330

Abstract

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We consider Lagrangian systems with Lagrange functions which exhibit a quadratic time dependence. We prove the existence of infinitely many solutions tending, as t → ± ∞ , to an «equilibrium at infinity». This result is applied to the Kirchhoff problem of a heavy rigid body moving through a boundless incompressible ideal fluid, which is at rest at infinity and has zero vorticity.

How to cite

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Bertotti, Maria Letizia, and Bolotin, Sergey V.. "Doubly asymptotic trajectories of Lagrangian systems and a problem by Kirchhoff." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 8.2 (1997): 93-100. <http://eudml.org/doc/244324>.

@article{Bertotti1997,
abstract = {We consider Lagrangian systems with Lagrange functions which exhibit a quadratic time dependence. We prove the existence of infinitely many solutions tending, as \( t \rightarrow \pm \infty \), to an «equilibrium at infinity». This result is applied to the Kirchhoff problem of a heavy rigid body moving through a boundless incompressible ideal fluid, which is at rest at infinity and has zero vorticity.},
author = {Bertotti, Maria Letizia, Bolotin, Sergey V.},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Lagrangian systems; Routh method; Doubly asymptotic trajectories; Calculus of variations; calculus of variations; Lagrangian functions; quadratic time dependence; existence of infinitely many solutions},
language = {eng},
month = {7},
number = {2},
pages = {93-100},
publisher = {Accademia Nazionale dei Lincei},
title = {Doubly asymptotic trajectories of Lagrangian systems and a problem by Kirchhoff},
url = {http://eudml.org/doc/244324},
volume = {8},
year = {1997},
}

TY - JOUR
AU - Bertotti, Maria Letizia
AU - Bolotin, Sergey V.
TI - Doubly asymptotic trajectories of Lagrangian systems and a problem by Kirchhoff
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1997/7//
PB - Accademia Nazionale dei Lincei
VL - 8
IS - 2
SP - 93
EP - 100
AB - We consider Lagrangian systems with Lagrange functions which exhibit a quadratic time dependence. We prove the existence of infinitely many solutions tending, as \( t \rightarrow \pm \infty \), to an «equilibrium at infinity». This result is applied to the Kirchhoff problem of a heavy rigid body moving through a boundless incompressible ideal fluid, which is at rest at infinity and has zero vorticity.
LA - eng
KW - Lagrangian systems; Routh method; Doubly asymptotic trajectories; Calculus of variations; calculus of variations; Lagrangian functions; quadratic time dependence; existence of infinitely many solutions
UR - http://eudml.org/doc/244324
ER -

References

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  2. BERTOTTI, M. L. - BOLOTIN, S. V., Doubly asymptotic trajectories of Lagrangian systems in homogeneous force fields. Annali di Matematica Pura e Applicata, in press. Zbl0971.70021
  3. BOLOTIN, S. V., The existence of homoclinic motions. Vestnik Moskov Univ., ser. I, Matem., Mekhan., 6, 1983, 98-103 (in Russian); English transl. in Moscow Univ. Math. Boll., 38, 1983, 117-123. Zbl0549.58019MR728558
  4. BOLOTIN, S. V. - KOZLOV, V. V., On the asymptotic solutions of the equations of dynamics. Vestnik Moskov Univ., ser. I, Matem., Mekhan., 4, 1980, 84-89 (in Russian); English transl. in Moscow Univ. Math. Boll., 35, 1980, 82-88. Zbl0439.70020MR585456
  5. CHAPLYGIN, S. A., On the motion of heavy bodies in an incompressible fluid, Collected papers 1. Izd-vo Akad. Nauk SSSR, Leningrad1933, 133-150 (in Russian). 
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  7. GIANNONI, F. - RABINOWTTZ, P. H., On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems. Nonlinear Differential Equations and Applications, 1, 1993, 1-46. Zbl0823.34050MR1273342DOI10.1007/BF01194038
  8. HAGEDORN, P., Über die Instabilität konservativer Systeme mit gyroskopischen Kräften. Arch. Rat. Mech. Anal., 58, 1975, 1-9. Zbl0329.70008MR395417
  9. KIRCHHOFF, G., Über die Bewegung eines Rotationskörpers in einer Flüssigkeit. J. für die reine und angewandte Mathematik, 71, 1870, 237-262. JFM02.0731.01
  10. KOZLOV, V. V., On the fall of an heavy rigid body in a ideal fluid. Meck. Tverd. Tela, 5, 1989, 10-17 (in Russian). Zbl1119.70009
  11. KOZLOV, V. V., On the stability of equilibrium positions in nonstationary force fields. J. Appl. Math. Mech., 55, 1991, 8-13. Zbl0747.70017MR1107493DOI10.1016/0021-8928(91)90054-X
  12. LAMB, H., Hydrodynamics. Dover Publications, New York1945. JFM36.0817.07
  13. PALAIS, R. S., Morse theory on Hilbert manifolds. Topology, 2, 1963, 299-340. Zbl0122.10702MR158410
  14. PALAIS, R. S. - SMALE, S., A generalized Morse theory. Bull. Amer. Math. Soc., 70, 1964, 165-171. Zbl0119.09201MR158411

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