Abstract
The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an example, that the indefiniteness of the energy does not forbid the stability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
AdlerS. L., Rev. Mod. Phys. 54, 729 (1982).
ArnoldV. I., Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
Boulware, D. G., In ahead of this time, in S. M. Christensen (ed.), B. S. De Witt: A Collection of Essays in Honour of his 60th Birthday, Bristol (1983).
BoulwareD. G. and DeserS., Phys. Rev. Lett. 55, 2656 (1984).
Ferraris, M., Francaviglia, M., and Magnano, G., Nonlinear gravitational Lagrangian, G.R.G., in print (1987).
FrancavigliaM. and KrupkaD. Ann. Inst. H. Poincaré 37, 295 (1982).
MoserJ., Comm. Pure Appl. Math. 11, 81 (1958).
MusickyD., J. Phys. A 11, 1 (1978).
NarnhoferH. and ThirringW., Phys. Lett. B 76, 428 (1978).
NekhoroshevN. N., Russian Math. Surveys 32, 1 (1978).
OstrogradskyM., Mem. Acad. St. Petersburg 6, 385 (1850).
PaisA. and UhlenbeckG. E., Phys. Rev. 79, 145 (1950).
Pagani, E., Tecchiolli, G., and Zerbini, S., Sulla stabita' dei sistemi lagrangiani di ordine superiore, Univ. Trento, unpublished report (1987).
PoincaréH., Les Méthodes nouvelles de la mécanique céleste, Guthier-Villars, Paris, 1982; Dover, New York, 1957.
SalvadoriL., Ann. Mat. Pura Appl. 69, 1 (1965).
SiegelC. L. and MoserJ. K., Lectures on Celestial Mechanics, Springer-Verlag, Berlin, Heidelberg, New York (1971).
StelleK. S., Phys. Rev. D16, 953 (1977).
Tecchiolli, G., Aspetti classici e quantistici di sistemi descritti da lagrangiane di ordine superiore nelle derivate, Tesi di Laurea in Fisica, Univ. di Trento (1986).
TkhaiV. N., P.M.M. U.S.S.R. 49, 273 (1985).
WhittakerE. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Univ. Press, Cambridge (1959).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pagani, E., Tecchiolli, G. & Zerbini, S. On the problem of stability for higher-order derivative Lagrangian systems. Lett Math Phys 14, 311–319 (1987). https://doi.org/10.1007/BF00402140
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00402140