Next: The generalized normal form Up: Normal forms Previous: Normal forms   Contents

## Lie transformations and the Birkhoff-Gustavson normal form

Consider an autonomous Hamiltonian system with degrees of freedom and a fixed point in the origin. We can always write the Hamiltonian as a formal power series in the phase space coordinates , , . With we have

Here is the -dimensional vector space of homogeneous polynomials of degree in variables, and we employ the multiindex notation

Note that the dimension of grows rapidly with (e.g. for we have ), such that any manipulation of for larger values of will have to be done by computer algebra rather than by hand''. We denote the space of all formal power series beginning with degree 2 by .

The Lie operator adjoint to a power series is the Poisson bracket of with some :

 (-1)

is a linear operator on for all . The Lie operator adjoint to the quadratic part of the Hamiltonian and restricted to the subspace is of central importance:

Note that maps monomials of degree to monomials of degree .

The Lie transformation associated with is the exponential of :

 (1)

Lie transformations are an adequate tool for Hamiltonian normal form theory because they are canonical [20].

Let be a Hamiltonian of type (1). is in Birkhoff-Gustavson normal form up to order if

 (2)

is in Birkhoff-Gustavson normal form if (6) holds for all . This definition is motivated by the fact that is an integral of motion if is in BGNF:

For any given Hamiltonian of type (1) we can proceed to the BGNF of in the following way: With some determine a new Hamiltonian by

 (3)

More explicitely we have

where stands for terms of order and higher. Collecting terms of equal order we get, using :

Assuming that is already in BGNF up to order , equation (8) allows several important conclusions: First of all, according to (8a) the contributions of order less than remain unchanged under the Lie transformation associated to . This shows that the transformed Hamiltonian is at least in normal form up to order , too. Secondly, (8b) indicates how to obtain a Hamiltonian which is in BGNF even up to order : From (8b) we get

 (3)

This homological equation [23] must be solved for and under the additional condition
 (4)

In other words, must be in the kernel (or null space) of :

Thus we have the following iterative process for the normalization of : For all we first solve the homological equation for the polynomials and and then obtain the remaining terms of the new Hamiltonian by evaluating (7). The calculation of the is a tedious but straight-forward task that can be left to computer algebra. The non-trivial key point is solving the homological equation.

Let us assume that the vector space can be decomposed into the direct sum of the kernel and range spaces of ,

 (5)

with . Then can uniquely be split into its kernel and range components

Hence, is uniquely determined by the kernel component of :
 (5)

Finally can be obtained by inverting
 (6)

Since there may be several preimages of under ( is uniquely determined by (14) up to any element of the null space of ) the normalization procedure is not unambiguous. However, we can always achieve unambiguity by additionally requiring to lie in the range space of .

The key point of the above procedure is the splitting (11). By means of the canonical transformation with

 (7)

Gustavson showed that for a Hamiltonian of type (1) equation (11) holds, since in the new coordinates the corresponding transformed operator is diagonal. Since yields the splitting (11), does as well. This proves the applicability of the BGNF theory to Hamiltonians of the Gustavson type (1): Every such Hamiltonian can, by means of a formal canonical transformation, be transformed into the equivalent Hamiltonian
 (8)

where and is in BGNF. The term formal'' indicates that we do not yet consider the convergence properties of the power series , and .

Next: The generalized normal form Up: Normal forms Previous: Normal forms   Contents
Martin_Engel 2000-05-25