  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.
