We now turn to the generic case where cannot be
decomposed into the direct sum of the kernel and range spaces of .
As a trivial example for the way in which this problem arises,
consider a particle with a single degree of freedom () which in
lowest order approximation is ``free'':
We
circumvent this problem by using Fredholm's alternative for :
In accordance with (18) it is natural to define
a new normal form.
Let be a Hamiltonian of type (1a) with an arbitrary
quadratic contribution . We say that is
in generalized normal form up to order if
Notice that (19) is not required to hold for -- in contrast to the corresponding definition (6) of the BGNF. The reason being that in general it is impossible to normalize , since transforming implies changing as well. For generic one has to expect . In Gustavson's case, however, (6) is always true for , because the Poisson bracket of with itself vanishes.
In order to
complete our definition of a normal form we have to specify the
explicit form of the scalar product.
For
and
we set [24,22]
We first write in yet another form. Linearizing Hamilton's
equations we obtain
the Hamiltonian matrix
, with the
-dimensional symplectic matrix
and the Hessian
.
Thus we have
In the form (22) can easily be used for determining the splitting (18) of . For the example (17) considered in the beginning of this section we obtain and therefore .
The method for transforming a given Hamiltonian into its generalized
normal form is exactly the same as the one described in the previous
section; one only has to replace (12b) by
(15) |
Note that for a Hamiltonian of Gustavson's type (1) the two definitions of normal form coincide, because in this case . So if is in BGNF (up to order ) it is in generalized normal form (up to order ), too. The utility of the normal form will become evident in the next section.