Cotangent bundle reduction and Poincaré–Birkhoff normal forms

•The construction of canonical coordinates for symmetry reduced Hamiltonian systems.•A detailed discussion of simple mechanical systems with symmetries.•An algorithm to compute the Poincaré–Birkhoff normal form of relative equilibria.•Example 1: a SO(3)SO(3) symmetry reduced three-body system.•Example 2: a SO(2)SO(2) symmetry reduced double spherical pendulum.

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.