118 research outputs found
On the Integrability of Tonelli Hamiltonians
In this article we discuss a weaker version of Liouville's theorem on the
integrability of Hamiltonian systems. We show that in the case of Tonelli
Hamiltonians the involution hypothesis on the integrals of motion can be
completely dropped and still interesting information on the dynamics of the
system can be deduced. Moreover, we prove that on the n-dimensional torus this
weaker condition implies classical integrability in the sense of Liouville. The
main idea of the proof consists in relating the existence of independent
integrals of motion of a Tonelli Hamiltonian to the size of its Mather and
Aubry sets. As a byproduct we point out the existence of non-trivial common
invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.Comment: 19 pages. Version accepted by Trans. Amer. Math. So
A variational approach to the study of the existence of invariant Lagrangian graphs
This paper surveys some results by the author and collaborators on the
existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The
presentation is based on an invited talk by the author at XIX Congresso Unione
Matematica Italiana (Bologna, 12-17 Sept. 2011).Comment: 28 page
Nearly circular domains which are integrable close to the boundary are ellipses
The Birkhoff conjecture says that the boundary of a strictly convex
integrable billiard table is necessarily an ellipse. In this article, we
consider a stronger notion of integrability, namely integrability close to the
boundary, and prove a local version of this conjecture: a small perturbation of
an ellipse of small eccentricity which preserves integrability near the
boundary, is itself an ellipse. This extends the result in [1], where
integrability was assumed on a larger set. In particular, it shows that (local)
integrability near the boundary implies global integrability. One of the
crucial ideas in the proof consists in analyzing Taylor expansion of the
corresponding action-angle coordinates with respect to the eccentricity
parameter, deriving and studying higher order conditions for the preservation
of integrable rational caustics.Comment: 64 pages, 3 figures. Final revised version, to appear on Geometric
and Functional Analysis (GAFA
Computing Mather's \beta-function for Birkhoff billiards
This article is concerned with the study of Mather's \beta-function
associated to Birkhoff billiards. This function corresponds to the minimal
average action of orbits with a prescribed rotation number and, from a
different perspective, it can be related to the maximal perimeter of periodic
orbits with a given rotation number, the so-called Marked length spectrum.
After having recalled its main properties and its relevance to the study of the
billiard dynamics, we stress its connections to some intriguing open questions:
Birkhoff conjecture and the isospectral rigidity of convex billiards. Both
these problems, in fact, can be conveniently translated into questions on this
function. This motivates our investigation aiming at understanding its main
features and properties. In particular, we provide an explicit representation
of the coefficients of its (formal) Taylor expansion at zero, only in terms of
the curvature of the boundary. In the case of integrable billiards, this result
provides a representation formula for the \beta-function near 0. Moreover, we
apply and check these results in the case of circular and elliptic billiards.Comment: 29 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1203.127
Lecture notes on Mather's theory for Lagrangian systems
These are introductory lecture notes on Mather's theory for Tonelli
Lagrangian and Hamiltonian systems. They are based on a series of lectures
given by the author at Universit\`a degli Studi di Napoli "Federico II" (April
2009), at University of Cambridge (academic year 2009-2010) and at Universitat
Polit\`ecnica de Catalunya (June 2010).Comment: 72 pages, 3 figure
Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms
In this article we prove that for a smooth fiberwise convex Hamiltonian, the
asymptotic Hofer distance from the identity gives a strict upper bound to the
value at 0 of Mather's function, thus providing a negative answer to a
question asked by K. Siburg in \cite{Siburg1998}. However, we show that
equality holds if one considers the asymptotic distance defined in
\cite{Viterbo1992}.Comment: 21pp, accepted for publication in Geometry & Topolog
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