Geometric mechanics, and symmetry: from finite to infinite dimensions ; with solutions to selcted exercises
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2009
|
| Series: | Oxford texts in applied and engineering mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVI, 515 S. Ill., graph. Darst. |
| ISBN: | 9780199212903 9780199212910 |
Staff View
MARC
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| 245 | 1 | 0 | |a Geometric mechanics, and symmetry |b from finite to infinite dimensions ; with solutions to selcted exercises |c Darryl D. Holm ; Tanya Schmah ; Cristina Stoica |
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2009 | |
| 300 | |a XVI, 515 S. |b Ill., graph. Darst. | ||
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| 490 | 0 | |a Oxford texts in applied and engineering mathematics | |
| 650 | 4 | |a Mechanics | |
| 650 | 4 | |a Geometry | |
| 650 | 4 | |a Symmetry (Mathematics) | |
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| 700 | 1 | |a Stoica, Cristina |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804140695843241984 |
|---|---|
| adam_text | Contents
Parti
1 Lagrangian
and Hamiitonian mechanics
3
1.1
Newtonian mechanics
1.2
Lagrangian mechanics
1.3
Constraints
1.4
The Legendre transform and Hamiitonian mechanics
1.5
Rigid bodies
3
13
18
24
30
2
Manifolds
43
2.1
Submanifolds of R
2.2
Tangent vectors and derivatives
2.3
Differentials and cotangent vectors
2.4
Matrix groups as submanifolds
2.5
Abstract manifolds
43
57
69
78
83
3
Geometry on manifolds
99
3.1
Vector fields
3.2
Differential
1
-forms
3.3
Tensors
3.4
Riemannian geometry
3.5
Symplectic geometry
99
112
117
128
139
4
Mechanics on manifolds
155
4.1
Lagrangian mechanics on manifolds
4.2
The Legendre transform and Hamilton s equations
4.3
Hamiitonian mechanics on
Poisson
manifolds
155
160
166
4.4
A brief look at symmetry, reduction and conserved quantities
175
xlv
Contents
5
Lie groups and Lie algebras
187
5.1
Matrix Lie groups and Lie algebras
187
5.2
Abstract Lie groups and Lie algebras
193
5.3
Isomorphisms of Lie groups and Lie algebras
199
5.4
The exponential map
203
6
Group actions, symmetries and reduction
209
6.1
Lie group actions
6.2
Actions of a Lie group on itself
6.3
Quotient spaces
6.4
Poisson
reduction
209
220
230
233
7
Euler-Poincaré
reduction: Rigid body and heavy top
241
7.1
Rigid body dynamics
7.2
Euler-Poincaré
reduction: the rigid body
7.3
Euler-Poincaré
reduction theorem
7.4
Modelling heavy-top dynamics
7.5
Euler-Poincaré
systems with advected parameters
241
248
255
261
270
8
Momentum maps
281
8.1
Definition and examples
8.2
Properties of momentum maps
281
291
9
Lie-Poisson reduction
295
9.1
The reduced Legendre transform
9.2
Lie-Poisson reduction: geometry
9.3
Lie-Poisson reduction: dynamics
9.4
Momentum maps revisited
9.5
Co-Adjoint orbits
9.6
Lie-Poisson brackets on semidirect products
296
301
307
310
315
318
10
Pseudo-rigid bodies
325
10.1
Modelling
10.2
Euler-Poincaré
reduction
325
330
Contents xv
10.3
Lie-Poisson
reduction
335
10.4
Momentum maps: angular momentum and circulation
337
Part
Π
351
11
EPDiff
353
11.1
Brief history of geometric ideal continuum motion
353
11.2
Geometric setting of ideal continuum motion
355
11.3
Euler-Poincaré
reduction for
continua
359
11.4
EPDiff:
Euler-Poincaré
equation on the diffeomorphisms
360
12
EPDiff solution behaviour
367
12.1
Introduction
367
12.2
Shallow-water background for peakons
371
12.3
Peakons and
puisons
378
13
Integrability of EPDiff in
1
D
385
13.1
The
CH
equation is bi-Hamiltonian
386
13.2
The CH equation is
isospectral
389
14
EPDiff in
n
dimensions
395
14.1
Singular momentum solutions of the EPDiff equation for
geodesic motion in higher dimensions
395
14.2
Singular solution momentum map Jsing
399
14.3
The geometry of the momentum map
406
14.4
Numerical simulations of EPDiff in two dimensions
410
15
Computational anatomy: contour matching using EPDiff
419
15.1
Introduction to computational anatomy (CA)
419
15.2
Mathematical formulation of template matching for CA
423
15.3
Outline matching and momentum measures
425
15.4
Numerical examples of outline matching
427
x¥¡
Contents
16
Computational anatomy:
Euler-Poincaré
image matching
433
16.1
Overview
433
16.2
Notation and Lagrangian formulation
434
16.3
Symmetry-reduced
Euler
equations
436
16.4
Euler-Poincaré
reduction
438
16.5
Semidirect-product examples
442
17
Continuum equations with advection
453
17.1
Kelvin-Stokes theorem for ideal fluids
453
17.2
Introduction to advected quantities
456
17.3
Euler-Poincaré
theorem
461
18
Euler-Poincaré
theorem for geophysical fluid dynamics
469
18.1
Kelvin circulation theorem for GFD
469
18.2
Approximate model fluid equations that preserve the
Euler-Poincaré
structure
476
18.3
Equations of 2D geophysical fluid motion
476
18.4
Equations of
3D
geophysical fluid motion
481
18.5
Variational principle for fluids in three dimensions
486
18.6
Euler s equations for a rotating stratified ideal
incompressible fluid
489
18.7
Well-posedness, ill-posedness, discretization and regularization
500
Bibliography
503
Index
509
Classical mechanics, one of the oldest branches of science, has undergone a
long evolution, developing hand in hand with many areas of mathematics, ;;
including calculus, differential geometry, and the theory of Lie groups and Lie
algebras. The modern formulations of Lagrangian and Hamiltonian mechanics,
in the coordinate-free language of differential geometry, are elegant and
general. They provide a unifying framework for many seemingly disparate
physical systems, such as N-particle systems, rigid bodies, fluids and other
continua,
and electromagnetic and quantum systems.
Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to
the geometric approach to classical mechanics, suitable for graduate students or
advanced undergraduates. It fills a gap between traditional classical mechanics
texts and advanced modern mathematical treatments of the subject.
The book begins by reviewing Lagrangian and Hamiltonian mechanics and
introducing appropriate elements of differential geometry and the theory of
Lie group actions. Thereafter, the main focus is on using symmetry to reduce
Hamilton s principle, leading to the
Euler-Poincaré
equations for dynamics on
Lie groups and semi-direct product spaces. Several applications are considered
in detail, including rigid and pseudo-rigid bodies, the heavy top, shallow water
waves, ideal fluids, geophysical fluid dynamics, and computational anatomy.
A variety of examples and figures illustrate the material, while the many
exercises, both solved and unsolved, make the book a valuable class text.
Cover photograph: iStockphoto
|
| any_adam_object | 1 |
| author | Holm, Darryl D. 1947- Schmah, Tanya Stoica, Cristina |
| author_GND | (DE-588)114074275 |
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| bvnumber | BV035767919 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)608532432 (DE-599)HBZHT015772082 |
| dewey-full | 531.01/516 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
| dewey-raw | 531.01/516 |
| dewey-search | 531.01/516 |
| dewey-sort | 3531.01 3516 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T22:04:04Z |
| institution | BVB |
| isbn | 9780199212903 9780199212910 |
| language | English |
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| physical | XVI, 515 S. Ill., graph. Darst. |
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| publisher | Oxford Univ. Press |
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| spelling | Holm, Darryl D. 1947- Verfasser (DE-588)114074275 aut Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises Darryl D. Holm ; Tanya Schmah ; Cristina Stoica Oxford [u.a.] Oxford Univ. Press 2009 XVI, 515 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford texts in applied and engineering mathematics Mechanics Geometry Symmetry (Mathematics) Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Schmah, Tanya Verfasser aut Stoica, Cristina Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627675&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627675&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Holm, Darryl D. 1947- Schmah, Tanya Stoica, Cristina Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises Mechanics Geometry Symmetry (Mathematics) Differentialgeometrie (DE-588)4012248-7 gnd Theoretische Mechanik (DE-588)4185100-6 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4185100-6 |
| title | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises |
| title_auth | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises |
| title_exact_search | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises |
| title_full | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises Darryl D. Holm ; Tanya Schmah ; Cristina Stoica |
| title_fullStr | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises Darryl D. Holm ; Tanya Schmah ; Cristina Stoica |
| title_full_unstemmed | Geometric mechanics, and symmetry from finite to infinite dimensions ; with solutions to selcted exercises Darryl D. Holm ; Tanya Schmah ; Cristina Stoica |
| title_short | Geometric mechanics, and symmetry |
| title_sort | geometric mechanics and symmetry from finite to infinite dimensions with solutions to selcted exercises |
| title_sub | from finite to infinite dimensions ; with solutions to selcted exercises |
| topic | Mechanics Geometry Symmetry (Mathematics) Differentialgeometrie (DE-588)4012248-7 gnd Theoretische Mechanik (DE-588)4185100-6 gnd |
| topic_facet | Mechanics Geometry Symmetry (Mathematics) Differentialgeometrie Theoretische Mechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627675&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627675&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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