A student's guide to Lagrangians and Hamiltonians:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2014
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | X, 173 S. graph. Darst. |
| ISBN: | 9781107042889 9781107617520 |
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MARC
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| 245 | 1 | 0 | |a A student's guide to Lagrangians and Hamiltonians |c Patrick Hamill |
| 246 | 1 | 3 | |a Lagrangians and Hamiltonians |
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Record in the Search Index
| _version_ | 1804152081494310912 |
|---|---|
| adam_text | Contents
R
Introduction page
ix
Acknowledgements
x
I LAGRANGIAN MECHANICS
1
Fundamental concepts
3
1.1
Kinematics
3
1.2
Generalized coordinates
5
1.3
Generalized velocity
7
1.4
Constraints
9
1.5
Virtual displacements
11
1.6
Virtual work and generalized force
12
1.7
Configuration space
13
1.8
Phase space
15
1.9
Dynamics
15
1.9.1
Newton s laws of motion
15
1.9.2
The equation of motion
16
1.9.3
Newton and Leibniz
16
1.10
Obtaining the equation of motion
18
1.10.1
The equation of motion in Newtonian mechanics
19
1.10.2
The equation of motion in Lagrangian mechanics
19
1.11
Conservation laws and symmetry principles
25
1.11.1
Generalized momentum and cyclic coordinates
27
1.11.2
The conservation of linear momentum
30
1.11.3
The conservation of angular momentum
33
1.11.4
The conservation of energy and the work function
36
1.12
Problems
41
vi
Contents
The calculus of variations
44
2.1
Introduction
44
2.2
Derivation of the Euler-Lagrange equation
45
2.2.1
The difference between
б
and
d
52
2.2.2
Alternate forms of the Euler-Lagrange equation
55
2.3
Generalization to several dependent variables
58
2.4
Constraints
60
2.4.1
Holonomic constraints
60
2.4.2
Non-holonomic constraints
64
2.5
Problems
67
Lagrangian dynamics
70
3.1
The principle of
ď
Alembert. A derivation of Lagrange s
equations
70
3.2
Hamilton s principle
73
3.3
Derivation of Lagrange s equations
75
3.4
Generalization to many coordinates
75
3.5
Constraints and Lagrange s
λ
-method
77
3.6
Non-holonomic constraints
81
3.7
Virtual work
83
3.7.1
Physical interpretation of the
Lagrange
multipliers
84
3.8
The
invariance
of the
Lagrange
equations
86
3.9
Problems
88
II HAMILTONIAN MECHANICS
91
Hamilton s equations
93
4.1
The Legendre transformation
93
4.1.1
Application to thermodynamics
95
4.2
Application to the Lagrangian. The Hamiltonian
97
4.3
Hamilton s canonical equations
98
4.4
Derivation of Hamilton s equations from Hamilton s
principle
100
4.5
Phase space and the phase fluid
101
4.6
Cyclic coordinates and the Routhian procedure
104
4.7
Symplectic notation
106
4.8
Problems
107
Canonical transformations;
Poisson
brackets
109
5.1
Integrating the equations of motion
109
5.2
Canonical transformations
110
Contents
vii
5.3
Poisson
brackets
117
5.4
The equations of motion in terms of
Poisson
brackets
119
5.4.1
Infinitesimal canonical transformations
120
5.4.2
Canonical invariants
124
5.4.3
Liouville s theorem
127
5.4.4
Angular momentum
128
5.5
Angular momentum in
Poisson
brackets
129
5.6
Problems
132
Hamilton-Jacobi theory
134
6.1
The Hamilton—Jacobi equation
135
6.2
The harmonic oscillator
-
an example
137
6.3
Interpretation of Hamilton s principal function
139
6.4
Relationship to
Schrödinger s
equation
140
6.5
Problems
142
Continuous systems
144
7.1
A string
144
7.2
Generalization to three dimensions
150
7.3
The Hamiltonian density
151
7.4
Another one-dimensional system
154
7.4.1
The limit of a continuous rod
156
7.4.2
The continuous Hamiltonian and the canonical
field equations
160
7.5
The electromagnetic field
162
7.6
Conclusion
166
7.7
Problems
166
Further reading
168
Answers to selected problems
169
Index
171
A Student s Guide to Lagrangians and Hamiltoniam
A concise but rigorous treatment of variational techniques, focusing primarily on
Lagrangian and Hamiltonian systems, this book is ideal for physics, engineering and
mathematics students.
The book begins by applying Lagrange s equations to a number of mechanical
systems. It introduces the concepts of generalized coordinates and generalized
momentum. Following this, the book turns to the calculus of variations to derive the
Euler-Lagrange equations. It introduces Hamilton s principle and uses this throughout
the book to derive further results. The Hamiltonian, Hamilton s equations, canonical
transformations,
Poisson
brackets and Hamilton-Jacobi theory are considered next.
The book concludes by discussing continuous Lagrangians and Hamiltonians and how
they are related to field theory.
Written in clear, simple language, and featuring numerous worked examples and
exercises to help students master the material, this book is a valuable supplement to
courses in mechanics.
PATRICK HAMILL is Professor Emeritus of Physics at San Jose State University.
He has taught physics for over
30
years, and his research interests are in celestial
mechanics and atmospheric physics.
|
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| author | Hamill, Patrick 1936- |
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| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| isbn | 9781107042889 9781107617520 |
| language | English |
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| spelling | Hamill, Patrick 1936- Verfasser (DE-588)106253137X aut A student's guide to Lagrangians and Hamiltonians Patrick Hamill Lagrangians and Hamiltonians 1. publ. Cambridge Cambridge Univ. Press 2014 X, 173 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Mechanics, Analytic Textbooks Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd rswk-swf Hamilton-Formalismus (DE-588)4376155-0 gnd rswk-swf Lagrange-Formalismus (DE-588)4316154-6 s DE-188 Hamilton-Formalismus (DE-588)4376155-0 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027218282&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027218282&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Hamill, Patrick 1936- A student's guide to Lagrangians and Hamiltonians Mechanics, Analytic Textbooks Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd Hamilton-Formalismus (DE-588)4376155-0 gnd |
| subject_GND | (DE-588)4316154-6 (DE-588)4376155-0 |
| title | A student's guide to Lagrangians and Hamiltonians |
| title_alt | Lagrangians and Hamiltonians |
| title_auth | A student's guide to Lagrangians and Hamiltonians |
| title_exact_search | A student's guide to Lagrangians and Hamiltonians |
| title_full | A student's guide to Lagrangians and Hamiltonians Patrick Hamill |
| title_fullStr | A student's guide to Lagrangians and Hamiltonians Patrick Hamill |
| title_full_unstemmed | A student's guide to Lagrangians and Hamiltonians Patrick Hamill |
| title_short | A student's guide to Lagrangians and Hamiltonians |
| title_sort | a student s guide to lagrangians and hamiltonians |
| topic | Mechanics, Analytic Textbooks Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd Hamilton-Formalismus (DE-588)4376155-0 gnd |
| topic_facet | Mechanics, Analytic Textbooks Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus Hamilton-Formalismus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027218282&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=027218282&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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