The uniqueness of normal forms via Lie transforms and its applications to Hamiltonian systems

A necessary and sufficient condition is given for the unique normal forms about critical elements-equilibrium points and periodic orbits. With some applications to the Birkhoff's normal form along with its generalized form by K.R. Meyer, the restricted problem of three bodies near L₄, the Birkohoff's normalization procedure, and the singular perturbation, of Hamiltonian systems are discussed.