1,720,993 research outputs found
A variational approach to the study of the existence of invariant lagrangian graphs
No full textThis paper surveys some recent results by the author and some collaborators, on the existence of invariant Lagrangian graphs for Tonelli Hamiltcmian systems
On the structure of action-minimizing sets for Lagrangian systems. (Princeton University Ph.D Thesis, PROQUEST LLC.)
No full textAction-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory
No full textJohn Mather’s seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry-Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather’s theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic.
Starting with the mathematical background from which Mather’s theory was born, Alfonso Sorrentino first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. He achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. Sorrentino then describes the whole theory and its subsequent developments and applications in their full generality.
Shedding new light on John Mather’s revolutionary ideas, this book is certain to become a foundational text in the modern study of Hamiltonian systems.
Alfonso Sorrentino is associate professor of mathematics at the University of Rome "Tor Vergata" in Italy. He holds a PhD in mathematics from Princeton University
On the Integrability of Tonelli Hamiltonians
Full text linkIn 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 Hamiltonian. © 2011 American Mathematical Society
Computing Mather's β-function for Birkhoff billiards
No full textThis article is concerned with the study of Mather's β-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 β-function near 0. Moreover, we apply and check these results in the case of circular and elliptic billiards
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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