UID:
almahu_9947367831202882
Format:
1 online resource (495 p.)
ISBN:
1-281-78859-7
,
9786611788599
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0-08-087269-7
Series Statement:
North-Holland mathematics studies ; 158
Content:
The differential geometric formulation of analytical mechanics not only offers a new insight into Mechanics, but also provides a more rigorous formulation of its physical content from a mathematical viewpoint.Topics covered in this volume include differential forms, the differential geometry of tangent and cotangent bundles, almost tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical aspects of variational and constraint theories.The book may be considered as a self-contained text and only p
Note:
Description based upon print version of record.
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Front Cover; Methods of Differential Geometry in Analytical Mechanics; Copyright Page; Contents; Preface; Chapter 1. Differential Geometry; 1.1 Some main results in Calculus on Rn; 1.2 Differentiable manifolds; 1.3 Differentiable mappings. Rank Theorem; 1.4 Partitions of unity; 1.5 Immersions and submanifold; 1.6 Submersions and quotient manifolds; 1.7 Tangent spaces. Vector fields; 1.8 Fibred manifolds. Vector bundles; 1.9 Tangent and cotangent bundles; 1.10 Tensor fields. The tensorial algebra. Riemannian metrics; 1.11 Differential forms. The exterior algebra; 1.12 Exterior differentiation
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1.13 Interior product1.14 The Lie derivative; 1.15 Distributions. Frobenius theorem; 1.16 Orientable manifolds. Integration. Stokes theorem; 1.17 de Rham cohomology. Poincare lemma; 1.18 Linear connections. Riemannian connections; 1.19 Lie groups; 1.20 Principal bundles. Frame bundles; 1.21 G-structures; 1.22 Exercises; Chapter 2. Almost tangent structures and tangent bundles; 2.1 Almost tangent structures on manifolds; 2.2 Examples. The canonical almost tangent structure of the tangent bundle; 2.3 Integrability; 2.4 Almost tangent connections
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2.5 Vertical and complete lifts of tensor fields to the tangent bundle2.6 Complete lifts of linear connections to the tangent bundle; 2.7 Horizontal lifts of tensor fields and connections; 2.8 Sasaki metric on the tangent bundle; 2.9 Affine bundles; 2.10 Integrable almost tangent structures which define fibrations; 2.11 Exercises; Chapter 3. Structures on manifolds; 3.1 Almost product structures; 3.2 Almost complex manifolds; 3.3 Almost complex connections; 3.4 Kahler manifolds; 3.5 Almost complex structures on tangent bundles (I); 3.6 Almost contact structures; 3.7 f-structures
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3.8 ExercisesChapter 4. Connections in tangent bundles; 4.1 Differential calculus on TM; 4.2 Homogeneous and semibasic forms; 4.3 Semisprays. Sprays. Potentials; 4.4 Connections in fibred manifolds; 4.5 Connections in tangent bundles; 4.6 Semisprays and connections; 4.7 Weak and strong torsion; 4.8 Decomposition theorem; 4.9 Curvature; 4.10 Almost complex structures on tangent bundles (II); 4.11 Connection in principal bundles; 4.12 Exercises; Chapter 5. Symplectic manifolds and cotangent bundles; 5.1 Symplectic vector spaces; 5.2 Symplectic manifolds; 5.3 The canonical symplectic structure
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5.4 Lifts of tensor fields to the cotangent bundle5.5 Almost product and almost complex structures; 5.6 Darboux Theorem; 5.7 Almost cotangent structures; 5.8 Integrable almost cotangent structures which define fibrations; 5.9 Exercises; Chapter 6. Hamiltonian systems; 6.1 Hamiltonian vector fields; 6.2 Poisson brackets; 6.3 First integrals; 6.4 Lagrangian submanifolds; 6.5 Poisson manifolds; 6.6 Generalized Liouville dynamics and Poisson brackets; 6.7 Contact manifolds and non-autonomous Hamiltonian systems; 6.8 Hamiltonian systems with constraints; 6.9 Exercises
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Chapter 7. Lagrangian systems
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English
Additional Edition:
ISBN 0-444-88017-8
Language:
English
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