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

## Normal forms

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:

${\displaystyle {\dot {x}}=F(x)}$

and ${\displaystyle x^{*}}$  a fixed point of the system: ${\displaystyle F(x^{*})=0}$ . Without lack of generality, we can assume ${\displaystyle x^{*}=0}$ . Assume that application ${\displaystyle F}$  can be develloped around ${\displaystyle 0}$ :

${\displaystyle F(x)=Ax+\dots }$

where the dots represent polynomial terms in ${\displaystyle x}$  of degree ${\displaystyle \geq 2}$ . There exists the following lema:

lemplo

Theorem:

Let ${\displaystyle h}$  be a vectorial polynom of order ${\displaystyle r\geq 2}$  and ${\displaystyle h(0)=h^{\prime }(0)=0}$ . The change of variable ${\displaystyle x=y+h(y)}$  transforms the differential equation ${\displaystyle {\dot {y}}=Ay}$  into the equation:

${\displaystyle {\dot {x}}=Ax+v(x)+\dots }$

where ${\displaystyle v(x)={\frac {\partial h}{\partial x}}Ax-Ah(x)}$  and where the dots represent terms of order ${\displaystyle >r}$ .

${\displaystyle {\dot {x}}=(I+\partial _{y}h)A(x-h(x))+\dots =Ax+[{\frac {\partial }{\partial x}}Ax-Ah(x)]}$

Note that ${\displaystyle {\frac {\partial h}{\partial x}}Ax-Ah(x)}$  is the Poisson crochet between ${\displaystyle Ax}$  and ${\displaystyle h(x)}$ . We note ${\displaystyle L_{A}h={\frac {\partial h}{\partial x}}Ax-Ah(x)}$  and we call the following equation:

${\displaystyle L_{A}h=v}$

the homological equation associated to the linear operator ${\displaystyle A}$ .

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 ${\displaystyle h}$  as a function of ${\displaystyle v}$  associated to the dynamics.

Let us call ${\displaystyle e_{i}}$ , ${\displaystyle i\in (1,\dots ,n)}$  the basis of eigenvectors of ${\displaystyle A}$ , ${\displaystyle \lambda _{i}}$  the associated eigenvalues, and ${\displaystyle x_{i}}$  the coordinates of the system is this basis. Let us write ${\displaystyle v=v_{r}+\dots }$  where ${\displaystyle v_{r}}$  contains the monoms of degree ${\displaystyle r}$ , that is the terms ${\displaystyle x^{m}=x_{1}^{m_{1}}\dots x_{n}^{m_{n}}}$ , ${\displaystyle m}$  being a set of positive integers ${\displaystyle (m_{1},\dots ,m_{n})}$  such that ${\displaystyle \sum m_{i}=r}$ . It can be easily checked (see [ma:equad:Arnold83]) that the monoms ${\displaystyle x^{m}e_{s}}$  are eigenvectors of ${\displaystyle L_{A}}$  with eigenvalue ${\displaystyle (m,\lambda )-\lambda _{s}}$  where ${\displaystyle (m,\lambda )=m_{1}\lambda _{1}+\dots +m_{n}\lambda _{n}}$ :

${\displaystyle L_{A}x^{m}e_{s}=[(m,\lambda )-\lambda _{s}]x^{m}e_{s}.}$

One can thus invert the homological equation to get a change of variable ${\displaystyle h}$  that eliminate the nonom considered. Note however, that one needs ${\displaystyle (m,\lambda )-\lambda _{s}\neq 0}$  to invert previous equation. If there exists a ${\displaystyle m=(m_{1},\dots ,m_{n})}$  with ${\displaystyle m_{i}\geq 0}$  and ${\displaystyle \sum m_{i}=r\geq 2}$  such that ${\displaystyle (m,\lambda )-\lambda _{s}=0}$ , then the set of eigenvalues ${\displaystyle \lambda }$  is called resonnant. If the set of eigenvalues is resonant, since there exist such ${\displaystyle m}$ , then monoms ${\displaystyle x^{m}e_{s}}$  can not be eliminated by a change of variable. This leads to the normal form theory ([ma:equad:Arnold83]).

## KAM theorem

An hamiltonian system is called integrable if there exist coordinates ${\displaystyle (I,\phi )}$  such that the Hamiltonian doesn't depend on the ${\displaystyle \phi }$ .

eqbasimom

${\displaystyle {\frac {dI_{i}}{dt}}=-{\frac {\partial H}{\partial \phi _{i}}}=0}$
${\displaystyle {\frac {d\phi _{i}}{dt}}=-{\frac {\partial H}{\partial I_{i}}}}$

Variables ${\displaystyle I}$  are called action and variables ${\displaystyle \phi }$  are called angles. Integration of equation eqbasimom is thus immediate and leads to:

${\displaystyle I=I_{0}}$

and ${\displaystyle \phi _{i}=\omega _{i}(I)t+\phi _{i}^{0}}$  where ${\displaystyle \omega _{i}(I)=-{\frac {\partial H}{\partial I_{i}}}}$  and ${\displaystyle \phi _{i}^{0}}$  are the initial conditions.

Let an integrable system described by an Hamiltonian ${\displaystyle H_{0}(I)}$  in the space phase of the action-angle variables ${\displaystyle (I,\phi )}$ . Let us perturb this system with a perturbation ${\displaystyle \epsilon H_{1}(I,\phi )}$ .

${\displaystyle H(I,\phi )=H_{0}(I)+\epsilon H_{1}(I,\phi )}$

where ${\displaystyle H_{1}}$  is periodic in ${\displaystyle \phi }$ .

If tori exist in this new system, there must exist new action-angle variable ${\displaystyle (I^{\prime },\phi ^{\prime })}$  such that:

eqdefHip

${\displaystyle H(I,\phi )=H^{\prime }(I^{\prime })}$

Change of variables in Hamiltonian system can be characterized ([ph:mecac:Goldstein80]) by a function ${\displaystyle S(\phi ,I^{\prime })}$  called generating function that satisfies:

${\displaystyle I={\frac {\partial S}{\partial \phi }}}$
${\displaystyle \phi ^{\prime }={\frac {\partial S}{\partial I^{\prime }}}}$

If ${\displaystyle S}$  admits an expension in powers of ${\displaystyle \epsilon }$  it must be:

${\displaystyle S=\phi I^{\prime }+\epsilon S_{1}(\phi ,I^{\prime })+\dots }$

Equation eqdefHip thus becomes:

equatfondKAM

${\displaystyle H_{0}(I^{\prime })+\epsilon \partial _{I_{i}^{\prime }}H(I^{\prime })\partial _{\phi _{i}}+H_{1}(I^{\prime },\phi )=H^{\prime }(I^{\prime })}$

Calling ${\displaystyle \omega _{0}}$  the frequencies of the unperturbed Hamiltionan ${\displaystyle H_{0}}$ :

${\displaystyle \omega _{0,i}(I^{\prime })=\partial _{i}H_{0}(I^{\prime })}$

Because ${\displaystyle H_{1}}$  and ${\displaystyle S_{1}}$  are periodic in ${\displaystyle \phi }$ , they can be decomposed in Fourier:

${\displaystyle H_{1}(I,\phi )=\sum _{m}H_{1,m}(I)e^{im\phi }}$
${\displaystyle S_{1}(I,\phi )=\sum _{m\neq 0}S_{1,m}(I)e^{im\phi }}$

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

${\displaystyle H^{\prime }(I^{\prime })=H_{0}^{\prime }(I^{\prime })+\epsilon H^{\prime }(I^{\prime })}$

and the relations:

${\displaystyle i.m.\omega _{0}(I^{\prime })S_{1,m}(I^{\prime })=H_{1,m}(I^{\prime })}$

Inverting formally previous equation leads to the generating function:

${\displaystyle S(\phi ,I^{\prime })=\phi I^{\prime }+\epsilon i\sum _{m\neq 0}{\frac {H_{1,m}(I^{\prime })}{m.\omega _{0}(I^{\prime })}}}$

The problem of the convergence of the sum and the expansion in ${\displaystyle \epsilon }$  has been solved by KAM. Clearly, if the ${\displaystyle \omega _{i}}$  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 ${\displaystyle \epsilon }$  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 ${\displaystyle \omega _{0,1}(I)}$  and ${\displaystyle \omega _{0,2}(I)}$  whose ratio ${\displaystyle \omega _{0,1}(I)/\omega _{0,2}(I)}$  is sufficiently irrational for the following relation to hold:
${\displaystyle \left|{\frac {\omega _{0,1}}{\omega _{0,2}}}-{\frac {r}{s}}\right|>{\frac {K(\epsilon )}{s^{2.5}}}{\mbox{ , for all integers }}r{\mbox{ and }}s,}$

where ${\displaystyle K}$  is a number that tends to zero with the ${\displaystyle \epsilon }$ .

2. To prove the convergence, KAM use an accelerated convergence method that, to calculate the torus at order ${\displaystyle n+1}$  uses the torus calculated at order ${\displaystyle n}$  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.