Sempre caro mi fu quest'ermo colle,
e questa siepe, che da tanta parte
dell'ultimo orizzonte il guardo esclude.
Ma sedendo e mirando, interminati
spazi di là da quella, e sovrumani
silenzi, e profondissima quiete
io nel pensier mi fingo; ove per poco
il cor non si spaura. E come il vento
odo stormir tra queste piante, io quello
infinito silenzio a questa voce
vo comparando; e mi sovvien l'eterno,
e le morte stagioni, e la presente
e viva, e il suon di lei. Così tra questa
immensità s'annega il pensier mio:
e il naufragar m'è dolce in questo mare.
(Giacomo Leopardi)

Normal form methods

with numerical applications


The lectures will be devoted to the study of the dynamics in a neighbourhood of an equilibrium point of differential equations, with particular emphasis on the case of a Hamiltonian system in a neighbourhood of an elliptic equilibrium.
The aim is to discuss at the same time the analytical methods and the numerical applications to the problem of stability.
Specific topics will be:
  1. Construction of first integrals for the Hamiltonian case, with quantitative estimates leading to exponential stability in Nekhoroshev's sense.
  2. Normal form methods based on the use of Lie series and Lie transform. The case of convergent series (Poincare'--Siegel center problem, Kolmogorov's theorem) and of exponential stability.
  3. Use of computer algebra in order to perform explicit expansions. Computer-assisted study of long time stability for specific models.
  4. Applications to the study of long time stability for realistic models, such as: the Lagrangian equilibria of the restricted problem of three bodies; the Lagrange theory of secular motions.

Participants will be encouraged and assisted in trying some applications to simple models, as introductory exercises.

Documentation on line:

Here you can find a copy of the slides that will be shown during the lectures.
Actually, the lectures will use only part of the material included here. However, we hope that the participants will appreciate the extra information.
The slides are written in CTL (Common Technical Language). It has some resemblance with english, so we hope it will be readable.
The documentation is subject to changes. Constructive criticism is appreciated.

  1. Documentation in size A4, portrait format, 10 point font, with space for personal notes on every page.
  2. Documentation in size A4, landscape format, 12 point font.

Documentation for exercises:

Suggested exercises include:
  1. Interactive exploration of the Poincaré section for a system of two harmonic oscillators with cubic nonlinearity (models of Contopoulos and of Hénon--Heiles).
  2. Using algebraic manipulation in order to calculate first integrals for systems of the type above
  3. Long time stability estimates with computer assisted methods.

Documentation and sorce code can be downloaded from the web page

Fairy tales do not tell children the dragons exist.
Children already know that dragons exist.
Fairy tales tell children the dragons can be killed.
(G.K. Chesterton)

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