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.
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:
- Construction of first integrals for the Hamiltonian case, with
quantitative estimates leading to exponential stability in
- 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
- Use of computer algebra in order to perform explicit
expansions. Computer-assisted study of long time stability for
- 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
Actually, the lectures will use only part of the material included
here. However, we hope that the participants will appreciate the
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.
in size A4, portrait format, 10 point font, with space for personal notes on
in size A4, landscape format, 12 point font.
Documentation for exercises:
Suggested exercises include:
- Interactive exploration of the Poincaré section for a system
of two harmonic oscillators with cubic nonlinearity (models of
Contopoulos and of Hénon--Heiles).
- Using algebraic manipulation in order to calculate first
integrals for systems of the type above
- Long time stability estimates with computer assisted methods.
Documentation and sorce code can be downloaded from the
Fairy tales do not tell children the dragons exist.|
Children already know that dragons exist.
Fairy tales tell children the dragons can be killed.
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