Invariant manifolds and periodic solutions of three degrees of freedom Hamiltonian systems
Hamiltonian systems considered near a stable equilibrium point can be analyzed using normalization techniques à la Birkhoff or, equivalently, averaging in one of its canonical forms. It is well known that two degrees of freedom systems become integrable upon normalization, the integrals being asymptotic integrals (valid for all time) for the original system. In the case of three degrees of freedom the situation is more complex: there are a number of results concerning integrability and there are many open problems. Both in two and three degrees of freedom, the periodic solutions admit systematic analysis although the complexity increases enormously with the dimension.
Most results are concerned with the generic cases but, keeping an eye on applications, we also have to allow for degeneracies and bifurcations arising from certain symmetry properties. As an illustration of some of the mathematical theory we shall consider applications in the theory of vibrations and in astrophysics.
KeywordsPeriodic Solution Periodic Orbit Hamiltonian System Invariant Manifold Discrete Symmetry
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