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A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows. (English) Zbl 1352.70046

Summary: We study the determination of the second-order normal form for perturbed Hamiltonians \(H_{\varepsilon }=H_0 +\varepsilon H_1 +\frac{\varepsilon ^2}{2} H_2\), relative to the periodic flow of the unperturbed Hamiltonian \(H_0\). The formalism presented here is global, and can be easily implemented in any computer algebra system. We illustrate it by means of two examples: the Hénon-Heiles and the elastic pendulum Hamiltonians.

MSC:

70H08 Nearly integrable Hamiltonian systems, KAM theory
70H05 Hamilton’s equations
70K45 Normal forms for nonlinear problems in mechanics
53D05 Symplectic manifolds (general theory)
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D22 Canonical transformations in symplectic and contact geometry

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