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Integrable Geodesic Flows on Two-Dimensional Surfaces

2000 Mathematics
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Release : 2000
Genre : Mathematics
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Book Synopsis Integrable Geodesic Flows on Two-Dimensional Surfaces by : A.V. Bolsinov

Download or read book Integrable Geodesic Flows on Two-Dimensional Surfaces written by A.V. Bolsinov. This book was released on 2000. Available in PDF, EPUB and Kindle. Book excerpt: From Moscow State University, Bolsinov (computer methods) and Fomenko (differential geometry) present a new approach to the qualitative analysis of the particular type of geodesic flow of Riemannian metrics on manifolds based on the theory of topological classification of integrable Hamiltonian systems. They begin by introducing the qualitative theory of integrable Hamiltonian systems, then discuss the class of integrable geodesic flows on two-dimensional surfaces from both the classical and contemporary perspectives. They classify the flows according to equivalence relations, such as isometry, the Liouville equivalence, the smooth and continuous trajectory equivalence, and the geodesic equivalence. They also explain the new technique that makes such classification possible. Many of their results have not been published before. The Russian original is Geometriia i topologiia integriruemykh geodezicheskikh potokov na poverkhnostiakhAnnotation copyrighted by Book News, Inc., Portland, OR

Integrable Geodesic Flows on Two-Dimensional Surfaces

2013-05-14 Mathematics
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Release : 2013-05-14
Genre : Mathematics
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Book Rating : 077/5 ( reviews)

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Book Synopsis Integrable Geodesic Flows on Two-Dimensional Surfaces by : A.V. Bolsinov

Download or read book Integrable Geodesic Flows on Two-Dimensional Surfaces written by A.V. Bolsinov. This book was released on 2013-05-14. Available in PDF, EPUB and Kindle. Book excerpt: Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.

Integrable Hamiltonian Systems

2004-02-25 Mathematics
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Release : 2004-02-25
Genre : Mathematics
Kind : eBook
Book Rating : 429/5 ( reviews)

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Book Synopsis Integrable Hamiltonian Systems by : A.V. Bolsinov

Download or read book Integrable Hamiltonian Systems written by A.V. Bolsinov. This book was released on 2004-02-25. Available in PDF, EPUB and Kindle. Book excerpt: Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors,

Integrability and Nonintegrability in Geometry and Mechanics

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
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Book Rating : 690/5 ( reviews)

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Book Synopsis Integrability and Nonintegrability in Geometry and Mechanics by : A.T. Fomenko

Download or read book Integrability and Nonintegrability in Geometry and Mechanics written by A.T. Fomenko. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . • 1111 Oulik'. n. . Chi" •. • ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Symplectic Geometry

1995-11-30 Mathematics
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Release : 1995-11-30
Genre : Mathematics
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Book Rating : 013/5 ( reviews)

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Book Synopsis Symplectic Geometry by : A.T. Fomenko

Download or read book Symplectic Geometry written by A.T. Fomenko. This book was released on 1995-11-30. Available in PDF, EPUB and Kindle. Book excerpt:

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