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## The generalized normal form

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'':

 (9)

takes on the form , such that we get and , and obviously

This shows that the normal form considered by Gustavson is not suitable for all types of .

We circumvent this problem by using Fredholm's alternative for :

 (10)

Here, as usual, the adjoint operator is defined via , where is any suitable scalar product. Below we will specify a particular scalar product that will simplify the following expressions as much as possible.

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

 (11)

is in generalized normal form if (19) holds for all .

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]

 (12)

where the bar denotes complex conjugation. It is easy to see that this product operation indeed has the properties of a scalar product. In [17] a somewhat different scalar product was introduced by choosing a special basis of and defining it to be orthonormal. These two scalar products are identical up to a normalization factor. However, the definition given here paves way for a more general approach and is much easier to use. This becomes apparent when trying to derive an explicit expression for . In [17] this was only achieved for a very restricted case, namely the so-called mirror machine'' or magnetic bottle Hamiltonians''. See section 3 for a discussion of this class of systems.

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

 (13)

with the abbreviation for . In order to find we rewrite its definition as

where we have used the relation which holds for any linear mapping on . Evaluating the time derivative yields , and we obtain as
 (14)

This expression is identical with (21) after conjugating and transposing .

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)

Since the splitting (18) holds for any , we have proven that any Hamiltonian can be normalized according to the generalized definition (19).

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.

Next: Quasi-integrals of motion Up: Normal forms Previous: Lie transformations and the   Contents
Martin_Engel 2000-05-25