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 assume $x^{*}=0$. 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 $L_{A}h={\frac {\partial h}{\partial x}}Ax-Ah(x)$ and we call the following equation:

$L_{A}h=v$

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 $x^{m}e_{s}$ are eigenvectors of $L_{A}$ with eigenvalue $(m,\lambda )-\lambda _{s}$ where $(m,\lambda )=m_{1}\lambda _{1}+\dots +m_{n}\lambda _{n}$:

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

One can thus invert the homological equation to get a change of variable $h$ 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 $x^{m}e_{s}$ 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 $(I,\phi )$ 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

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

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

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

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

are not destroyed: The series converges.