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

From Wikibooks, open books for an open world
Jump to: navigation, search

Normal forms[edit]

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[edit]

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.