Unique Normal Forms Research Articles

Unique normal forms for vector fields and Hamiltonians

On construit des formes canoniques pour tous les champs vectoriels (resp. hamiltoniens) en equilibre a 0∈R n relativement au groupe des diffeomorphismes formels (resp. des transformations canoniques) tangentiels a l'identite

Unique normal forms for lambda calculus with surjective pairing

We consider the equational theory λπ of λ-calculus extended with constants π, π 0 , π 1 and axioms for surjective pairing: π 0 ( πXY ) = X , π 1 ( πXY ) = Y , π ( π 0 X )( π 1 X ) = X . Two reduction systems yielding the equality of λπ are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a unique normal form. Some additional properties and problems in the syntactical analysis of λπ and the corresponding reduction systems are discussed.

Unique normal forms for planar vector fields

Soit X un germe d'un champ vectoriel analytique (ou formel) en equilibre a O∈R 2 avec des valeurs propres purement imaginaires ±[a 0 i, a 0 >0. Alors X est formellement equivalent a precisement un des champs vectoriels polynomiaux suivant (ou x=(ρ) 1/2 cosφ, y=(ρ) 1/2 sin φ, pour des coordonnees appropriees (x, y)):(#7B-A 0 ): a 0 ∂/∂φ; (#7B-A k , ±): (a 0 ±[ρ k )∂/∂φ; (#7B-B j , ±): (a 0 +a 1 ρ+...+a j ρ j )∂/∂φ±ρ j (1+bρ j )∂/∂ρ, j>0

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.