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 :

Note that maps monomials of degree to monomials of degree .

The *Lie transformation* associated with is the exponential
of
:

(1) |

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

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

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

(4) |

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 ,

Hence, is uniquely determined by the kernel component of :

(5) |

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

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