Geodesic
On Stability in Hamiltonian Systems with Two Degrees of Freedom
Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 202-208.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than $1$. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.
Keywords: Hamiltonian system with two degrees of freedom, oscillator, Lyapunov stability, KAM theory, Poincaré mapping.
Mots-clés : equilibrium position, equilibrium position 
@article{MZM_2014_95_2_a2,
     author = {Yu. N. Bibikov},
     title = {On {Stability} in {Hamiltonian} {Systems} with {Two} {Degrees} of {Freedom}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {202--208},
     publisher = {mathdoc},
     volume = {95},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a2/}
}
TY  - JOUR
AU  - Yu. N. Bibikov
TI  - On Stability in Hamiltonian Systems with Two Degrees of Freedom
JO  - Matematičeskie zametki
PY  - 2014
SP  - 202
EP  - 208
VL  - 95
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a2/
LA  - ru
ID  - MZM_2014_95_2_a2
ER  - 
%0 Journal Article
%A Yu. N. Bibikov
%T On Stability in Hamiltonian Systems with Two Degrees of Freedom
%J Matematičeskie zametki
%D 2014
%P 202-208
%V 95
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a2/
%G ru
%F MZM_2014_95_2_a2
Yu. N. Bibikov. On Stability in Hamiltonian Systems with Two Degrees of Freedom. Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 202-208. http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a2/

[1] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR | Zbl

[2] A. G. Sokolskii, “Ob ustoichivosti avtonomnoi gamiltonovoi sistemy s dvumya stepenyami svobody pri rezonanse pervogo poryadka”, PMM, 41:1 (1977), 24–33 | MR | Zbl

[3] A. D. Bryuno, “Ob ustoichivosti v sisteme Gamiltona”, Matem. zametki, 40:3 (1986), 385–392 | MR | Zbl

[4] A. M. Lyapunov, “Issledovanie odnogo iz osobennykh sluchaev zadachi ob ustoichivosti dvizheniya”, Sobranie sochinenii, T. 2, Izd-vo AN SSSR, M.–L., 1956, 272–331

[5] Yu. K. Mozer, “O razlozhenii uslovno-periodicheskikh dvizhenii v skhodyaschiesya stepennye ryady”, UMN, 24:2 (1969), 165–211 | MR

[6] Yu. N. Bibikov, Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, Izd-vo Leningr. un-ta, L., 1991 | MR | Zbl

[7] J. K. Moser, Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., 81, Amer. Math. Soc., Providence, RI, 1968 | MR | Zbl