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

Normal forms
As written in  it is very powerfull not to solve differential equations but to tranform them into a simpler differential equation. Let the system:

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 $$0$$:

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

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:

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 ) that the monoms $$x^me_s$$ are eigenvectors of $$L_A$$ with eigenvalue $$(m,\lambda)-\lambda_s$$ where $$(m,\lambda)=m_1\lambda_1+\dots+m_n\lambda_n$$:

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.

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

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

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 $$\epsilon H_1(I,\phi)$$.

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:

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

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

Equation eqdefHip thus becomes:

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

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

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

and the relations:

Inverting formally previous equation leads to the generating function:

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.