# 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:

$\dot x=F(x)$

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

$F(x)=Ax+\dots$

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

lemplo

Theorem:

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

$\dot x=Ax+v(x)+\dots$

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

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

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

$L_Ah=v$

the homological equation associated to the linear operator $A$.

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 $h$ as a function of $v$ associated to the dynamics.

Let us call $e_i$, $i\in (1,\dots,n)$ the basis of eigenvectors of $A$, $\lambda_i$ the associated eigenvalues, and $x_i$ the coordinates of the system is this basis. Let us write $v=v_r+\dots$ where $v_r$ contains the monoms of degree $r$, that is the terms $x^{m}=x_1^{m_1}\dots x_n^{m_n}$, $m$ being a set of positive integers $(m_1,\dots,m_n)$ such that $\sum m_i=r$. It can be easily checked (see [ma:equad:Arnold83]) that the monoms $x^me_s$ are eigenvectors of $L_A$ with eigenvalue $(m,\lambda)-\lambda_s$ where $(m,\lambda)=m_1\lambda_1+\dots+m_n\lambda_n$:

$L_Ax^me_s=[(m,\lambda)-\lambda_s]x^me_s.$

One can thus invert the homological equation to get a change of variable $h$ that eliminate the nonom considered. Note however, that one needs $(m,\lambda)-\lambda_s\neq 0$ to invert previous equation. If there exists a $m=(m_1,\dots,m_n)$ with $m_i\geq 0$ and $\sum m_i=r\geq 2$ such that $(m,\lambda)-\lambda_s=0$, then the set of eigenvalues $\lambda$ is called resonnant. If the set of eigenvalues is resonant, since there exist such $m$, then monoms $x^me_s$ 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 $(I,\phi)$ such that the Hamiltonian doesn't depend on the $\phi$.

eqbasimom

$\frac{dI_i}{dt}=-\frac{\partial H}{\partial \phi_i}=0$

$\frac{d\phi_i}{dt}=-\frac{\partial H}{\partial I_i}$

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

$I=I_0$

and $\phi_i=\omega_i(I)t+\phi_i^0$ where $\omega_i(I)=-\frac{\partial H}{\partial I_i}$ and $\phi_i^0$ are the initial conditions.

Let an integrable system described by an Hamiltonian $H_0(I)$ in the space phase of the action-angle variables $(I,\phi)$. Let us perturb this system with a perturbation $\epsilon H_1(I,\phi)$.

$H(I,\phi)=H_0(I)+\epsilon H_1(I,\phi)$

where $H_1$ is periodic in $\phi$.

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

eqdefHip

$H(I,\phi)=H^\prime(I^\prime)$

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

$I=\frac{\partial S}{\partial \phi}$

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

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

$S=\phi I^\prime+\epsilon S_1(\phi,I^\prime)+\dots$

Equation eqdefHip thus becomes:

equatfondKAM

$H_0(I^\prime)+\epsilon\partial_{I^\prime_i}H(I^\prime) \partial_{\phi_i} + H_1(I^\prime,\phi)=H^\prime(I^\prime)$

Calling $\omega_0$ the frequencies of the unperturbed Hamiltionan $H_0$:

$\omega_{0,i}(I^\prime)=\partial_iH_0(I^\prime)$

Because $H_1$ and $S_1$ are periodic in $\phi$, they can be decomposed in Fourier:

$H_1(I,\phi)=\sum_m H_{1,m}(I)e^{im\phi}$

$S_1(I,\phi)=\sum_{m\neq 0} S_{1,m}(I)e^{im\phi}$

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

$H^\prime(I^\prime)=H_0^\prime(I^\prime)+\epsilon H^\prime(I^\prime)$

and the relations:

$i.m.\omega_0(I^\prime) S_{1,m}(I^\prime)=H_{1,m}(I^\prime)$

Inverting formally previous equation leads to the generating function:

$S(\phi,I^\prime)=\phi I^\prime+\epsilon i \sum_{m\neq 0}\frac{H_{1,m}(I^\prime)}{m.\omega_0(I^\prime)}$

The problem of the convergence of the sum and the expansion in $\epsilon$ has been solved by KAM. Clearly, if the $\omega_i$ 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 $\epsilon$ 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 $\omega_{0,1}(I)$ and $\omega_{0,2}(I)$ whose ratio $\omega_{0,1}(I)/\omega_{0,2}(I)$ is sufficiently irrational for the following relation to hold:

$\left|\frac{\omega_{0,1}}{\omega_{0,2}}-\frac{r}{s}\right|> \frac{K(\epsilon)}{s^{2.5}}\mbox{ , for all integers } r\mbox{ and } s,$

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

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