The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards.
The geometry of Calogero-Moser systems
We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the -th power of the elliptic curve, where is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for...
The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)
Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions,...
The geometry of nondegeneracy conditions in completely integrable systems
The -graded symplectic Floer cohomology of monotone Lagrangian submanifolds.
The hamiltonian formalism in higher order variational problems
The Helmholtz conditions for the difference equations systems.
The Invariance of Poincaré's Generating Function for Canonical Transformations.
The inverse problem for flat kinetic minus potencial Lagrangians.
The inverse problem in the calculus of variations: new developments
We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for . We then examine a new class of solutions in arbitrary dimension and give some non-trivial examples in dimension 3.
The Jacobi variational principle revisited
The Kolmogorov-Arnold-Moser theorem.
The Kowalevski's top and the method of Syzygies
A new algorithm for finding separation coordinates is tested on the example of Kowalev ski’s top.
The Lagrangian and Hamiltonian formulations for the waves in a compressible fluid with the Hall current.
In questo lavoro si ricavano: 1) l'equazione d'onda linearizzata, 2) la formulazione Lagrangiana, 3) la formulazione Hamiltoniana, nella teoria della propagazione ondosa in un fluido comprimibile descritto dalle equazioni della magnetofluidodinamica ideale in presenza di corrente Hall.
The Legendre transformation
The magnetic geodesic flow in a homogeneous field on the complex projective space.
The -centre problem of celestial mechanics for large energies
We consider the classical three-dimensional motion in a potential which is the sum of attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the centres, we find a universal behaviour for all energies above a positive threshold. Whereas for there are no bounded orbits, and for there is just one closed orbit, for the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set....
The obstacle problem and direct products of unfurled swallowtails
The quantum stability problem for time-periodic perturbations of the harmonic oscillator
The reduction of symplectic structure and Sternberg construction