Nekhoroshev Theory

Glossary

Quasi integrable Hamiltonian:

A Hamiltonian is quasi integrable if it is close to another Hamiltonian whose associated system is integrable by quadrature. Here, we will consider real analytic Hamiltonians on a domain $\mathcal{D}$ which admit a holomorphic extension on a complex strip ${\mathcal{D}}_{\mathrm {\mathbb{C}}}$ around $\mathcal{D}$ and the closeness to the integrable Hamiltonian is measured with the supremum norm ‖.‖_∞ over ${\mathcal{D}}_{\mathrm {\mathbb{C}}}$.

Exponentially stable Hamiltonian:

An integrable Hamiltonian governs a system which admits a collection of first integral. We say that an integrable Hamiltonian h is exponentially stable if for any small enough Hamiltonian perturbation of h, the solutions of the perturbed system are at least defined over timescales which are exponentially long with respect to the inverse of the size of the perturbation. Moreover, the first integrals of the integrable system should remain nearly constant...

Appendix A

An Example of Divergence Without Small Denominators

We would like to develop the following example of Neishtadt (Nekhorochev 1977) where the phenomenons of divergence without small denominators and summation up to an exponentially small remainder appear in its simplest setting (see also Benettin 2005 for another example).

Consider the quasi integrable system governed by the Hamiltonian

$$ {\displaystyle \begin{array}{l}\mathcal{H}\left({I}_1,{I}_2;{\theta}_1,{\theta}_2\right)={I}_1-\varepsilon \left[{I}_2-\cos \left({\theta}_1\right)f\left({\theta}_2\right)\right]\\ {}\mathrm {defined}\ \mathrm {over}\ {\mathrm {\mathbb{R}}}^2\times {\mathbbm{T}}^2,\end{array}} $$

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with

$$ f\left(\theta \right)=\sum \limits_{m\ge 1}\frac{\alpha_m}{m}\cos \left(m\, \theta \right), $$

where α_m = e^−αm for 0 < α ≤ 1, this last choice corresponds to the exponential decay of the Fourier coefficients for a holomorphic function.

Indeed, f^′(θ) =  Im (g(−α + iθ)) where g(z) = (exp(z) − 1)⁻¹ and the complex pole of f which is closest to the real axis is located at −iα.

Conversely, all real function which admits a holomorphic extension over a complex strip of width α (i.e.: ∣ Im (z) ∣  ≤ α) has Fourier coefficients bounded by ${\left(C{\alpha}_m\right)}_{m\in {\mathrm {\mathbb{N}}}^{\ast }}$ for some constant C > 0.

As in the section “Hamiltonian Perturbation Theory”, it is possible to eliminate formally completely the fast angle θ₁ in the perturbation without the occurrence of any small denominators.

Indeed, one can consider a normalizing transformation generated by X(θ₁, θ₂) = ∑_n ≥ 1εⁿX_n(θ₁, θ₂) where the functions X_n satisfy the homological equations:

$$ {\displaystyle \begin{array}{c}{\partial}_{\theta_1}{X}_1\left({\theta}_1,{\theta}_2\right)=\cos \left({\theta}_1\right)f\left({\theta}_2\right)\ \mathrm {and}\ \\ {}{\partial}_{\theta_1}{X}_n\left({\theta}_1,{\theta}_2\right)={\partial}_{\theta_2}{X}_{n-1}\left({\theta}_1,{\theta}_2\right).\end{array}} $$

The solutions can be written ${X}_n\left({\theta}_1,{\theta}_2\right)=\cos \left({\theta}_1-n\frac{\pi }{2}\right){f}^{\left(n-1\right)}\left({\theta}_2\right)$, hence:

$$ {\displaystyle \begin{array}{l}X\left({\theta}_1,{\theta}_2\right)=\sum \limits_{n\in {\mathrm {\mathbb{N}}}^{\ast }}{\varepsilon}^n\sum \limits_{m\in {\mathrm {\mathbb{N}}}^{\ast }}\frac{\alpha_m}{m^2}{m}^n\\ {}\qquad \cdot \cos \left({\theta}_1-n\frac{\pi }{2}\right)\sin \left(m{\theta}_2+n\frac{\pi }{2}\right)\end{array}} $$

and, for instance,

$$ X\left(\frac{\pi }{2},0\right)=\sum \limits_{m\in {\mathrm {\mathbb{N}}}^{\ast }}\sum \limits_{k\in \mathrm {\mathbb{N}}}\frac{{\mathrm {e}}^{-\alpha m}}{m^2}{\left(\varepsilon m\right)}^{2k+1} $$

which is divergent for all ε > 0 since ε ≥ 1/m for m large enough.

We see that divergence comes from coefficients arising with successive differentiations.

On the other hand, if one considers the transformation generated by the truncated Hamiltonian ${\sum}_{n\ge 1}^N{\varepsilon}^n{X}_n\left({\theta}_1,{\theta}_2\right)$ then the transformed Hamiltonian becomes

$$ \mathcal{H}\left({I}_1,{I}_2;{\theta}_1,{\theta}_2\right)={I}_1-\varepsilon {I}_2+{\varepsilon}^{N+1}{\partial}_{\theta_2}{X}_N\left({\theta}_1,{\theta}_2\right), $$

where ${\partial}_{\theta_2}{X}_N\left({\theta}_1,{\theta}_2\right)=\cos \left({\theta}_1-N\frac{\pi }{2}\right){f}^{(N)}\left({\theta}_2\right)$.

Especially, with g(z) = 1/ exp (z) − 1, we have:

$$ {\displaystyle \begin{array}{l}{\partial}_{\theta_2}{X}_{2N}\left({\theta}_1,{\theta}_2\right)\\ {}\quad =-\cos \left({\theta}_1\right)\operatorname{Re}\left({g}^{\left(2N-1\right)}\left(-\alpha +i{\theta}_2\right)\right)\\ {}\mathrm {for}\ N>0,\\ {}{\partial}_{\theta_2}{X}_{2N+1}\left({\theta}_1,{\theta}_2\right)=\sin \left({\theta}_1\right)\operatorname{Im}\left({g}^{(2N)}\left(-\alpha +i{\theta}_2\right)\right)\\ {}\mathrm {for}\ N>0,\end{array}} $$

Now, around the real axis, the main term in the asymptotic expansion of g⁽ⁿ⁾(z) as n goes to infinity is given by the derivative of the polar term 1/z in the Laurent expansion of g at 0.

Indeed, we consider the function h(z) = g(z) − 1/z which is real analytic and admits an analyticity width of size 2π around the real axis. Hence, Cauchy estimates ensure that for n large enough and z in the strip of width p around the real axis, the derivative h⁽ⁿ⁾(z) becomes negligible with respect to the derivative of 1/z.

Consequently, g⁽ⁿ⁾(z) is equivalent to (−1)ⁿn ! /z^n + 1 as n goes to infinity for z in the strip of width p around the real axis.

Hence, the remainder ${\partial}_{\theta_2}{X}_N\left({\theta}_1,{\theta}_2\right)$ has a size of order (N − 1) ! /α^N for N large. This latter estimate cannot be improved, since ${\partial}_{\theta_2}{X}_{2N}\left(\pi, 0\right)$ and ${\partial}_{\theta_2}{X}_{2N+1}\left(\pi /2,\alpha \tan \left(\pi /4N+2\right)\right)$ admit the same size of order (N − 1) ! /α^N for N large.

Now, we can make a “summation at the smallest term” as in section “The Case of a Single Frequency System” to obtain an optimal normalization with an exponentially small remainder.