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Gauge Mechanicsкнига
- Авторы: Mangiarotti L., Sardanashvily G.
- Год издания: 1998
- Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
- Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
- Объём: 355 страниц
- ISBN: 981-02-3603-4
- Аннотация: Preface This book presents in a unified way modern geometric methods in analytical mechanics, based on the application of jet manifolds and connections. As is well known, the technique of Poisson and symplectic spaces provide the adequate Hamiltonian formulation of conservative mechanics. This formulation, however, cannot be extended to time-dependent mechanics subject to time-dependent transformations. We will formulate non-relativistic time-dependent mechanics as a particular field theory on fibre bundles over a time axis. The geometric approach to field theory is based on the identification of classical fields with sections of fibred manifolds. Jet manifolds provide the adequate mathematical language for Lagrangian field theory, while the Hamiltonian one is phrased in terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian field theory leads us in a straightforward manner to Lagrangian time-dependent mechanics. At the same time, the canonical polysymplectic form on a momentum phase space of time-dependent mechanics reduces to the canonical exterior 3-form which plays the role similar to a symplectic form in conservative mechanics. With this canonical 3-form, we introduce the canonical Poisson structure and formulate Hamiltonian time-dependent mechanics in terms of Hamiltonian connections and Hamiltonian forms. Note that the theory of non-linear differential operators and the calculus of variations are conventionally phrased in terms of jet manifolds. On the other hand, jet formalism provides the contemporary language of differential geometry to deal with non-linear connections, represented by sections of jet bundles. Only jet spaces enable us to treat connections, Lagrangian and Hamiltonian dynamics simultaneously. In fact, the concept of connection is the main link throughout the book. Connections on a configuration space of time-dependent mechanics are reference frames. Holonomic connections on a velocity phase space define non-relativistic dynamic equations which are also related to other types of connections, and can be written as non-relativistic geodesic equations. Hamiltonian time-dependent mechanics deals with Hamiltonian connections whose geodesics are solutions of the Hamilton equations. The presence of a reference frame, expressed in terms of connections, is the main peculiarity of time-dependent mechanics. In particular, each reference frame defines an energy function, and quantizations with respect to different reference frames are not equivalent. Another important peculiarity is that a Hamiltonian fails to be a scalar function under time-dependent transformations. As a consequence, many well-known constructions of conservative mechanics fail to be valid for time-dependent mechanics, and one should follow methods of field theory. At the same time, there is the essential difference between field theory and time-dependent mechanics. In contrast with gauge potentials in field theory, connections on a configuration space of time-dependent mechanics fail to be dynamic variables since their curvature vanishes identically. Following geometric methods of field theory, we obtain the frame-covariant formulation of time-dependent mechanics. By analogy with gauge field theory, one may speak about gauge time-dependent mechanics. In comparison with non-relativistic time-dependent mechanics, a configuration space of relativistic mechanics does not imply any preferable fibration over a time. To construct the velocity phase space of relativistic mechanics, we therefore use formalism of jets of submanifolds. At the same time, Hamiltonian relativistic mechanics is seen as an autonomous Hamiltonian system on the constraint space of relativistic hyperboloids. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the convenience of the reader, several mathematical facts and notions are included as an Interlude, thus making our exposition self-contained. Contents. Preface. Introduction. 1 Interlude: bundles, jets, connections. 1.1 Fibre bundles. 1.2 Multivector fields and differential forms. 1.3 Jet manifolds. 1.4 Connections. 1.5 Bundles with symmetries. 1.6 Composite fibre bundles. 2 Geometry of Poisson manifolds. 2.1 Jacobi structure. 2.2 Contact structure. 2.3 Poisson structure. 2.4 Symplectic structure. 2.5 Presymplectic structure. 2.6 Reduction of symplectic and Poisson structures. 2.7 Appendix. Poisson homology and cohomology. 2.8 Appendix. More brackets. 2.9 Appendix. Multisymplectic structures. 3 Hamiltonian systems. 3.1 Dynamic equations. 3.2 Poisson Hamiltonian systems. 3.3 Symplectic Hamiltonian systems. 3.4 Presymplectic Hamiltonian systems. 3.5 Dirac Hamiltonian systems. 3.6 Dirac constraint systems. 3.7 Hamiltonian systems with symmetries. 3.8 Appendix. Hamiltonian field theory. 4 Lagrangian time-dependent mechanics. 4.1 Fibre bundles over R. 4.2 Dynamic equations. 4.3 Dynamic connections. 4.4 Non-relativistic geodesic equations. 4.5 Reference frames. 4.6 Free motion equations. 4.7 Relative acceleration. 4.8 Lagrangian systems. 4.9 Newtonian systems. 4.10 Holonomic constraints. 4.11 Non-holonomic constraints. 4.12 Lagrangian conservation laws. 5 Hamiltonian time-dependent mechanics. 5.1 Canonical Poisson structure. 5.2 Hamiltonian connections and Hamiltonian forms. 5.3 Canonical transformations. 5.4 The evolution equation. 5.5 Degenerate systems. 5.6 Quadratic degenerate systems. 5.7 Hamiltonian conservation laws. 5.8 Time-dependent systems with symmetries. 5.9 Systems with time-dependent parameters. 5.10 Unified Lagrangian and Hamiltonian formalism. 5.11 Vertical extension of Hamiltonian formalism. 5.12 Appendix. Time-reparametrized mechanics. 6 Relativistic mechanics. 6.1 Jets of submanifolds. 6.2 Relativistic velocity and momentum phase spaces. 6.3 Relativistic dynamics. 6.4 Relativistic geodesic equations. 7 Appendix A. Geometry of BRST mechanics. 8 Appendix B. On quantum time-dependent mechanics. Bibliography.
- Добавил в систему: Сарданашвили Геннадий Александрович
Прикрепленные файлы
| № | Имя | Описание | Имя файла | Размер | Добавлен |
|---|---|---|---|---|---|
| 1. | Полный текст | scanner copy | Gauge_Mechanics.pdf | 39,9 МБ | 3 мая 2013 [sardanashvily] |
| 2. | Полный текст | book98.pdf | 2,8 МБ | 31 мая 2012 [sardanashvily] |