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## Quasi-integrals of motion

By construction, for a Hamiltonian in BGNF is a formal integral of motion (see section 2.1). We now show how to find an analogous formal integral for a Hamiltonian in generalized normal form. Our results are similar to the findings of Meyer and Hall [19], but the proof differs in some details. We have tried to make the exposition as transparent as possible by focusing on just those aspects that are essential for the reasoning.

We write as

 (16)

and decompose by means of the Jordan-Chevalley decomposition [25] into its diagonalizable and nilpotent parts and :
 (17)

Existence and uniqueness of this decomposition are assured by the Jordan normal form theorem for matrices. Define the diagonalizable component and the nilpotent component of by
 (18)

such that . We are now in the position to prove the main

Theorem: For a Hamiltonian in generalized normal form the diagonalizable part of is a formal integral of motion.

For the proof we must show that the Poisson bracket of with vanishes for all . We start with and then proceed to the case .

By virtue of Jacobi's identity for the Poisson bracket we have

 (19)

This expression is zero if and commute. In order to show that the latter is the case we first remark that the matrices , and are infinitesimally symplectic [26], i.e. they satisfy (for ). Direct computation shows that for an infinitesimally symplectic matrix the Lie operator adjoint to the quadratic polynomial can be written as . Thus we have for the Lie operators and adjoint to and :
 (20)

It is one of the key advantages of this formulation of the theory that we can characterize all the important operators , and which operate in a space of the high dimension by matrices of the considerably smaller dimension : , and .

We now show that for two commuting matrices , the corresponding Lie operators , (defined as above) commute as well:

 (21)

Because this implies that the right hand side of (27) is zero, and thus .

For we proceed in the following way: We show that diagonalizability and nilpotence of the matrices and carry over to the corresponding Lie operators and ; these properties then imply that the null spaces of and coincide and that , which in turn means .

Consider a unitary matrix that transforms into the diagonal matrix . Inserting twice the identity into the expression for we get

With , and denoting in the new coordinates by , we obtain
 (22)

Application of this transformed operator to any of the basis monomials of yields, because is diagonal, an eigenvalue equation with the eigenfunction and a certain eigenvalue -- thus diagonalizability of is shown.

Now consider any . is a polynomial in of degree less than or equal to , since by nilpotence there is some such that . This polynomial is related to in the following way:

Iterating this expression and evaluating for we get
 (23)

which implies nilpotence of , because (31) holds for all .

Identity of the null spaces of and is a direct implication of diagonalizability: Application of the diagonalized operator (cf. (30)) yields

So the eigenspaces corresponding to the eigenvalue 0 of and are identical.

Finally, we determine how acts on polynomials in . Notice that is nilpotent, because its adjoint is. With we obtain for any :

because and commute (since the corresponding matrices and commute; cf. (29)). is zero for and because is in generalized normal form. From this it follows for that

here, for the sake of notational convenience, we have again turned to the coordinates as defined above. Linear independence of the basis monomials then gives the result and thus

and we have proven the theorem.

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