Geodesic
On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian
Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 343-352.

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

A time-periodic system with one degree of freedom is investigated. The system is assumed to have an equilibrium position, in the vicinity of which the Hamiltonian is represented as a convergent series.This series does not contain members of the second degree, whereas the members to some finite degree $\ell$ do not depend explicitly on time. The algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian in members to degree $\ell$, inclusive. As an application, a special case is considered when the expansion of the Hamiltonian begins with members of the third degree. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.
Keywords: Hamiltonian system, canonical transformation, stability. 
@article{ND_2015_11_2_a9,
     author = {A. P. Markeev},
     title = {On the {Birkhoff} transformation in the case of complete degeneracy of the quadratic part of the {Hamiltonian}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {343--352},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2015_11_2_a9/}
}
TY  - JOUR
AU  - A. P. Markeev
TI  - On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian
JO  - Russian journal of nonlinear dynamics
PY  - 2015
SP  - 343
EP  - 352
VL  - 11
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2015_11_2_a9/
LA  - ru
ID  - ND_2015_11_2_a9
ER  - 
%0 Journal Article
%A A. P. Markeev
%T On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian
%J Russian journal of nonlinear dynamics
%D 2015
%P 343-352
%V 11
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2015_11_2_a9/
%G ru
%F ND_2015_11_2_a9
A. P. Markeev. On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian. Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 343-352. http://geodesic.mathdoc.fr/item/ND_2015_11_2_a9/

[1] Birkgof D., Dinamicheskie sistemy, UdGU, Izhevsk, 1999, 408 pp.

[2] Dzhakalya G. E. O., Metody teorii vozmuschenii dlya nelineinykh sistem, Nauka, Moskva, 1979, 319 pp.

[3] Markeev A. P., “Uproschenie struktury form tretei i chetvertoi stepenei v razlozhenii funktsii Gamiltona pri pomoschi lineinogo preobrazovaniya”, Nelineinaya dinamika, 10:4 (2014), 447–464

[4] Kozlov V. V., Furta S. D., Asimptotiki reshenii silno nelineinykh sistem differentsialnykh uravnenii, MGU, Moskva, 1996, 244 pp.

[5] Malkin I. G., Teoriya ustoichivosti dvizheniya, Nauka, Moskva, 1966, 532 pp.

[6] Markeev A. P., Teoreticheskaya mekhanika, NITs «Regulyarnaya i khaoticheskaya dinamika», Moskva – Izhevsk, 2007, 592 pp.