Cotangent bundle reduction and Poincar\'e-Birkhoff normal forms
  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 Poincar{\'e}-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 Poincar{\'e}-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. 
\"Unver \c{C}ift\c{c}i and Holger Waalkens and Henk Broer
math.DS math-ph math.MP nlin.CD