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^[1]

are not destroyed: The series converges^[2].

↑ 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 $\epsilon$ .
↑ 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.

[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.

[1]

[2]