Abstract
We consider a conservative Hamiltonian system with two degrees of freedom where the linearized system consists of two harmonic oscillators whose frequenciesω 1 andω 2 are commensurable or close to being so. Near the equilibrium of such a system we prove the existence of natural families of periodic orbits and analyze their relation to the natural families which follow from Lyapunov's theorem. Some of these families have been established in recent years whereas others have only been discussed in a formal manner. We give a complete description of all families and provide an easy existence proof at the same time.
The main application is to the restricted problem of three bodies nearL 4 for mass ratios nearμ q/p whereqω 1 =pω 2. Our only restriction is thatq andp are relatively prime. In particular our results allow a complete discussion of the behavior of the natural families nearL 4 as μ passes throughμ q/p . The results agree with numerical findings of Deprit and others.
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References
Deprit, A. and Deprit-Bartholomé, A.: 1967,Astron. J. 72, 173–179.
Deprit, A. and Henrard, J.: 1969, in Giacaglia (ed.),Periodic Orbits, Stability and Resonances, Reidel, 1–18.
Hale, J. K.: 1969,Ordinary Differential Equations, Wiley-Interscience, New York.
Henrard, J.: 1970,Celes. Mech. 1, 437–466.
Meyer, K. R. and Palmore, J. I.: 1970,J. Differential Equations 8, 264–276.
Roels, J.: 1971,J. Differential Equations 9, 300–324.
Schmidt, D. S. and Sweet, D.: 1973,J. Differential Equations 14, 597–609.
Sweet, D.: 1973,J. Differential Equations 14, 171–183.
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Schmidt, D.S. Periodic solutions near a resonant equilibrium of a Hamiltonian system. Celestial Mechanics 9, 81–103 (1974). https://doi.org/10.1007/BF01236166
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DOI: https://doi.org/10.1007/BF01236166