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Cotangent bundle reduction and Poincaré-Birkhoff normal forms. (English) Zbl 1417.37192

Summary: In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
53D20 Momentum maps; symplectic reduction
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