236 research outputs found
Il problema degli n-corpi in relatività generale
No full textTraduzione dal francese all’italiano dell’ultimo lavoro di Tullio Levi Civita pubblicato postumo nel 1950, “Le problème des n corps en relativité générale”. Introduzione, traduzione e note a cura di Franco Cardin, Sara Di Ruzza e Leonardo Donà
Symplectic rigidity and weak commutativity
No full textWe present a new and simple proof of Eliashberg–Gromov’s theorem based on the notion of C0-commutativity introduced by Cardin and Viterbo
ON VISCOSITY AND GEOMETRICAL SOLUTIONS OF HAMILTON JACOBI EQUATIONS
No full textIn this paper the author relates a geometric solution for the Cauchy problem to a convex Hamilton-Jacobi equation with its unique viscosity solution. The viscosity solution is shown to correspond to the inf sup of the Morse family generating the geometric solution given previously by the author [Nuovo Cimento B (11) 104 (1989), no. 5, 525--544].
Reviewed by George Kossiori
Global Finite Generating Functions for Field Theory
No full textWe introduce an infinite-dimensional version of the Amann-Conley-Zehnder reduction for a class of boundary problems related to nonlinear perturbed elliptic operators with symmetric derivative. We construct global generating functions with finite auxiliary parameters, describing the solutions as critical points in a finite-dimensional space
The global finite structure of generic envelope loci for Hamilton-Jacobi equations
No full textWe discuss in some detail the existence of global generating functions describing Lagrangian submanifolds connected with evolution problems for Hamilton–Jacobi H–J equations. First, we produce a physical application of a result by Viterbo: for generic in a suitable sense Hamiltonian functions and initial data, the envelopes, i.e., the wave front sets, related to Hamilton–Jacobi problems are globally finitely generated. Furthermore, we show how to compute global space–time generating functions with finite parameters for geometric solutions of a H–J equation of the evolution kind
Global World Functions
No full textStarting from the Amann-Conley-Zehnder finite reduction framework in the non-compact Viterbo’s version, we discuss the existence of global generat- ing function with a finite number of auxiliary parameters describing the two-points Characteristic Relation related to the geodesic problem in the Hamiltonian formal- ism. This applies both to Analytical Mechanics and to General Relativity - we construct a global object generalizing the World Function introduced by Synge, which is well-defined only locally. Whenever the auxiliary parameters can be fully removed, Synge’s World Function is restored
Global Hamilton principal functions of the eikonal equations on and
No full textThis monograph grew out of a series of lectures given at the XXVI Summer School of Mathematical Physics, Ravello, September 2001, organized by G.N.F.M. (gruppo Nazionale di Fisica Matematica) of I.N.d.A.M. (Istituto Nazionale di Alta Matematica, Roma), at the Department of Mathematics of the University of Torino in the academic years 2000/2001 and 2001/2002, and at the Departmento of Physical Sciences of the University of Napoli, May 2003.
The elements of Symplectic Geometry and Analytical Mechanics on which these lectures are based can be found in the literature of the seventies and eighties of the last century. The bibliography is of course far from complete and refers the reader to some of the important contributions. Here, we introduce only the essential notions of symplectic geometry needed for application to the geometrical theory of the Hamilton-Jacobi equation and to the control theory of static systems. Most of these notions are well known, but the way they are assembled and used is new in many respects.
A fundamental role in the present approach is played by the notion of generating family and by two operations: the composition of generating families of symplectic relations and the canonical lift from objects on manifolds (submanifolds, relations, mappings, vector fields, etc.) to symplectic objects on the corresponding cotangent bundles. Generating families describe special subsets of cotangent bundles which we call Lagrangian sets. A Lagrangian set is a Lagrangian submanifold (which may be immersed) if the generating family is a Morse family. However, there are physically interesting examples of Lagrangian sets which are not Lagrangian submanifolds. An advantage of considering generating families as fundamental objects is that, while the composition of two symplectic relations may not be a smooth relation, the composition of two generating families is always a smooth function. In other words, the symplectic creed as formulated by A. Weinstein in his article Symplectic geometry (1981) in the form everything is a Lagrangian submanifold, which means that one should try to express objects in symplectic geometry and mechanics in terms of Lagrangian submanifolds, is here replaced by everything has a generating family.
The geometrical theory of the Hamilton-Jacobi equation is closely related to Geometrical Optics. The symplectic formulation of Hamiltonian Optics presented here differs from other formulations illustrated in papers and well known reference books cited in the Bibliography and it is, in my opinion, very close to the original ideas of Hamilton. From a geometrical view-point a Hamilton-Jacobi equation is a coisotropic submanifold of a cotangent bundle. A geometrical solution is a Lagrangian set described by a generating family and contained in the coisotropic submanifold. There are two fundametal symplectic relations associated with a Hamilton-Jacobi equation, the characteristic relation and the characteristic reduction. The two corresponding generating families are the Hamilton principal function and the complete solution of the Hamilton-Jacobi equation, respectively. By composing the latter with its transpose we get the former. Since the characteristic relation is a singular Lagrangian submanifold, the Hamilton principal function is necessarily a generating family and not a two-point function as in the classical theory. Cauchy data (or sources of systems of rays), mirror and lenses are represented by symplectic relations thus, by generating families. Then the Cauchy problem and the actions of a lens or of a mirror on a system of rays are translated into the composition of generating families.
What is presented here is only a first approach to Geometrical Optics based on the notions of symplectic relation and generating family. We do not cover many important examples of optical phenomena, which can be found in standard reference books (e.g. Synge, Luneburg, Buchdahl) and which probably can be treated within this framework.
Perhaps, the use of generating families and symplectic relations does not yield a revolutionary progress in Hamiltonian Optics, but we are obliged to introduce these concepts if, for example, we want to give a global meaning to the Hamilton characteristic function, as shown in Chapters 3 and 4.
Symplectic relations and generating families can play an interesting role also in the control theory of static systems, including thermostatic systems. Chapter 5 is devoted to this matter. Our approach is based on the notion of control relation and on an extended version of the virtual work principle for constrained systems with non-controlled degrees of freedom (hidden variables). Several examples of singular phenomena concerning static systems and thermostatics are illustrated. In particular, it is shown how the Maxwell rule follows as a theorem from the extended virtual work principle. Thermostatics of simple and composite systems is here described in the four-dimensional state space, with global coordinates (S, V, P, T), entropy, volume, pressure, absolute temperature, endowed with the natural symplectic structure induced by the first principle of thermodynamics.
An outline of the basic tools of calculus on manifolds needed in our discussion is given in Appendix A. A supplementary note (Appendix B) written in collaboration with Franco Cardin (Dipartimento di Matematica Pura e Applicata, Università di Padova), is devoted to the calculus of global principal Hamilton functions for the eikonal equations on the two-dimensional sphere S2 and pseudo-sphere H2
Elementary symplectic topology and mechanics
No full textThis is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects
Trasporto ottimo, sistemi viventi
No full textIn questa rassegna si traccia un incontro con la teoria del trasporto ottimo, fornendo alcune notizie sulla sua nascita, sulla sua rinnovata riscoperta e conseguenti inattese nuove applicazioni. Si parler\`a di trasporto ottimo per il sistema arterioso, della legge di Kleiber, delle radici degli alberi, e infine, della diffusione della muffa Physarum Polycephalum e delle sue caratteristiche geodetiche. Il filo conduttore delle sezioni, apparentemente scollegate, \`e in realt\`a sempre il problema di Monge-Kantorovich, a partire dalla sua lettura originale statica, passando per una lettura dinamica di tipo stazionario, fino ad una lettura dinamica di tipo non stazionario (nel caso delle muffe), tendente asintoticamente nel tempo a Monge-Kantorovich
ON TOPOLOGICAL DEGREE AND POINCARE' DUALITY
No full textIn this note we investigate about some relations between Poincar\'e
dual and other topological objects, such as intersection index, topological degree, and Maslov index of
Lagrangian submanifolds. A simple proof of the Poincar\'e-Hopf theorem is
recalled. The Lagrangian submanifolds are the geometrical, multi-valued,
solutions of physical problems of evolution governed by Hamilton-Jacobi
equations: the computation of the algebraic number of the branches is showed to
be performed by using Poincar\'e dual
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