We show how to apply to Hamiltonian differential systems recent results for studying the periodic orbits of a differential system using the averaging theory. We have chosen two classical integrable Hamiltonian systems, one with the Hooke potential and the other with the Kepler potential, and we study the periodic orbits which bifurcate from the periodic orbits of these integrable systems, first perturbing the Hooke Hamiltonian with a nonautonomous potential, and second perturbing the Kepler problem with an autonomous potential.

Topics
Astrodynamics, Hamiltonian mechanics, Central-force, Coordinate system, Recurrence relations, Power series, Vector calculus
1.
Arnold
,
V. I.
,
Kozlov
,
V. V.
, and
Neishtadt
,
A. I.
,
Mathematical Aspects of Classical and Celestial Mechanics
(
Springer-Verlag
,
Berlin
,
1997
).
Google Scholar
 
2.
Abraham
,
R.
 and
Marsden
,
J. E.
,
Foundations of Mechanics
, 2nd ed. (
Benjamin/Cummings
,
Reading, MS
,
1978
).
Google Scholar
 
3.
Buicǎ
,
A.
,
Françoise
,
J. P.
, and
Llibre
,
J.
, “
Periodic solutions of nonlinear periodic differential systems with a small parameter
,”
Commun. Pure Appl. Anal.
 
6
,
103
(
2007
).
Google Scholar
Crossref
Search ADS
 
4.
Goldstein
,
H.
,
Poole
,
C.
, and
Safko
,
J.
,
Classical Mechanics
, 3rd ed. (
Addison-Wesley
,
Reading
,
2002
).
Google Scholar
 
5.
Greenspan
,
B. D.
 and
Holmes
,
P. J.
, “
Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators
,”
SIAM J. Math. Anal.
 
15
,
69
(
1984
).
https://doi.org/10.1137/0515003
Google Scholar
Crossref
Search ADS
 
6.
Guckenheimer
,
J.
 and
Holmes
,
P.
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector fields
(
Springer
,
New York
,
1983
).
Google Scholar
Crossref
Search ADS
 
7.
Llibre
,
J.
,
Teixeira
,
M. A.
, and
Torregrosa
,
J.
, “
Limit cycles bifurcating from a k-dimensional isochronous set center contained in Rn with k≤n
,”
Math. Phys., Anal. Geom.
 
10
,
237
(
2007
).
https://doi.org/10.1007/s11040-007-9030-7
Google Scholar
Crossref
Search ADS
 
8.
Lochak
,
P.
 and
Meunier
,
C.
,
Applied Mathematical Sciences
Vol.
72
, (
Springer-Verlag
,
New York
,
1988
).
Google Scholar
 
9.
Meyer
,
K. R.
,
Hall
,
G. R.
, and
Offin
,
D.
, 2nd ed. 
Applied Mathematical Science
, Vol.
90
, (
Springer-Verlag
,
New York
,
2009
).
Google Scholar
 
10.
Sanders
,
J. A.
 and
Verhulst
,
F.
,
Applied Mathematical Science
, Vol.
59
, (
Springer-Verlag
,
New York
,
1985
).
Google Scholar
 
11.
Verhulst
,
F.
,
Nonlinear Differential Equations and Dynamical Systems
(
Springer-Verlag
,
Berlin
,
1996
).
Google Scholar
Crossref
Search ADS
 
© 2010 American Institute of Physics.
2010
American Institute of Physics
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