Introduction to Mathematical Physics/Some mathematical problems and their solution/Use of change of variables

Normal formsEdit

As written in ([ma:equad:Arnold83]) it is very powerfull not to solve differential equations but to tranform them into a simpler differential equation. ([ma:equad:Arnold83],[ma:equad:Guckenheimer83],[ma:equad:Berry78]) Let the system:

 

and   a fixed point of the system:  . Without lack of generality, we can assume  . Assume that application   can be develloped around  :

 

where the dots represent polynomial terms in   of degree  . There exists the following lema:

lemplo

Theorem:

Let   be a vectorial polynom of order   and  . The change of variable   transforms the differential equation   into the equation:

 

where   and where the dots represent terms of order  .

 

Note that   is the Poisson crochet between   and  . We note   and we call the following equation:

 

the homological equation associated to the linear operator  .

We are now interested in the reverse step of theorem lemplo: We have a nonlinear system and want to find a change of variable that transforms it into a linear system. For this we need to solve the homological equation, {\it i.e.} to express   as a function of   associated to the dynamics.

Let us call  ,   the basis of eigenvectors of  ,   the associated eigenvalues, and   the coordinates of the system is this basis. Let us write   where   contains the monoms of degree  , that is the terms  ,   being a set of positive integers   such that  . It can be easily checked (see [ma:equad:Arnold83]) that the monoms   are eigenvectors of   with eigenvalue   where  :

 

One can thus invert the homological equation to get a change of variable   that eliminate the nonom considered. Note however, that one needs   to invert previous equation. If there exists a   with   and   such that  , then the set of eigenvalues   is called resonnant. If the set of eigenvalues is resonant, since there exist such  , then monoms   can not be eliminated by a change of variable. This leads to the normal form theory ([ma:equad:Arnold83]).

KAM theoremEdit

An hamiltonian system is called integrable if there exist coordinates   such that the Hamiltonian doesn't depend on the  .

eqbasimom

 

 

Variables   are called action and variables   are called angles. Integration of equation eqbasimom is thus immediate and leads to:

 

and   where   and   are the initial conditions.


Let an integrable system described by an Hamiltonian   in the space phase of the action-angle variables  . Let us perturb this system with a perturbation  .

 

where   is periodic in  .

If tori exist in this new system, there must exist new action-angle variable   such that:

eqdefHip

 

Change of variables in Hamiltonian system can be characterized ([ph:mecac:Goldstein80]) by a function   called generating function that satisfies:

 

 

If   admits an expension in powers of   it must be:

 

Equation eqdefHip thus becomes:

equatfondKAM

 

Calling   the frequencies of the unperturbed Hamiltionan  :

 

Because   and   are periodic in  , they can be decomposed in Fourier:

 

 

Projecting on the Fourier basis equation equatfondKAM one gets the expression of the new Hamiltonian:

 

and the relations:

 

Inverting formally previous equation leads to the generating function:

 

The problem of the convergence of the sum and the expansion in   has been solved by KAM. Clearly, if the   are resonnant (or commensurable), the serie diverges and the torus is destroyed. However for non resonant frequencies, the denominator term can be very large and the expansion in   may diverge. This is the {\bf small denominator problem}.

In fact, the KAM theorem states that tori with ``sufficiently incommensurable frequencies[1]

are not destroyed: The series converges[2].

  1. In the case two dimensional case the KAM theorem proves that the tori that are not destroyed are those with two frequencies   and   whose ratio   is sufficiently irrational for the following relation to hold:

     

    where   is a number that tends to zero with the  .

  2. To prove the convergence, KAM use an accelerated convergence method that, to calculate the torus at order   uses the torus calculated at order   instead of the torus at order zero like an classical Taylor expansion. See ([ma:equad:Berry78]) for a good analogy with the relative speed of the Taylor expansion and the Newton's method to calculate zeros of functions.