Normal form

Henk W. Broer. Normal forms in perturbation theory. In Robert A. Meyers (eds.) Encyclopedia of Complexity and Systems Science. ' (2009). MathSciNet, arXiv.

The method of normal forms transforms the Hamiltonian of an equation via a canonical transformation to remove (or attenuate) non-resonant portions of the nonlinearity, replacing them with more tractable terms. For instance, normal forms can replace a quadratic nonlinearity with a cubic one. They are particularly useful in nonlinear wave equations.

In Bo-p2 the method of normal forms was shown to be compatible with the I-method, and used to improve the low-regularity global regularity theory for certain nonlinear Schrodinger equations.

Normal forms should not be confused with the unrelated concept of a null form. They achieve a similar effect as gauge transformations, although the latter arise from the differential geometry of connections and bundles rather than from the structure of the Hamiltonian.

We use normal forms in order to simplify the Taylor series of a Hamiltonian system at an equilibrium point so that we can then apply perturbation theory. The lower order truncation of the system may be better understood than the full system, and thus we can view the system as a perturbation of some truncation which is in a tractable form. With the Birkhoff normal form, we can view the system as a perturbation of a free Hamiltonian plus terms that commute with the free Hamiltonian.