Numerical Hamiltonian problems:
The purpose of this book is to present a unified introduction to the interdisciplinary and growing field of symplectic integration of Hamiltonian problems. These problems appear frequently in physics and other sciences and recent investigations suggest that they should be simulated by symplectic int...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London u.a.
Chapman & Hall
1994
|
| Edition: | 1. ed. |
| Series: | Applied mathematics and mathematical computation
7 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | The purpose of this book is to present a unified introduction to the interdisciplinary and growing field of symplectic integration of Hamiltonian problems. These problems appear frequently in physics and other sciences and recent investigations suggest that they should be simulated by symplectic integrators, i.e. by numerical methods that preserve the symplectic structure of phase space, thus reproducing the main feature of Hamiltonian dynamics The authors explain in detail how to derive, analyse and use symplectic integrators. Virtually all currently available symplectic methods are reported using a coherent notation. Numerical Hamiltonian Problems includes five preliminary chapters on Hamiltonian problems and numerical methods, so that very little background knowledge should be required. Numerical examples are provided throughout |
| Physical Description: | XII, 207 S. graph. Darst. |
| ISBN: | 0412542900 9780486824109 |
Staff View
MARC
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| 490 | 1 | |a Applied mathematics and mathematical computation |v 7 | |
| 520 | 3 | |a The purpose of this book is to present a unified introduction to the interdisciplinary and growing field of symplectic integration of Hamiltonian problems. These problems appear frequently in physics and other sciences and recent investigations suggest that they should be simulated by symplectic integrators, i.e. by numerical methods that preserve the symplectic structure of phase space, thus reproducing the main feature of Hamiltonian dynamics | |
| 520 | |a The authors explain in detail how to derive, analyse and use symplectic integrators. Virtually all currently available symplectic methods are reported using a coherent notation. Numerical Hamiltonian Problems includes five preliminary chapters on Hamiltonian problems and numerical methods, so that very little background knowledge should be required. Numerical examples are provided throughout | ||
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| 650 | 7 | |a système hamiltonien |2 inriac | |
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| adam_text | Contents
Preface xi
1 Hamiltonian systems 1
1.1 Hamiltonian systems 1
1.2 Examples of Hamiltonian systems 3
1.2.1 The harmonic oscillator 3
1.2.2 The pendulum 3
1.2.3 The double harmonic oscillator 5
1.2.4 Kepler s problem 6
1.2.5 A modified Kepler problem 8
1.2.6 H6non Heiles Hamiltonian 12
2 Symplecticness 15
2.1 The solution operator 15
2.2 Preservation of area 16
2.2.1 Concept of preservation of area 16
2.2.2 Preservation of area and dynamics 18
2.2.3 Preservation of area as a characteristic property 19
2.3 Checking preservation of area: Jacobians 19
2.4 Checking preservation of area: differential forms 20
2.5 Symplectic transformations 21
2.6 Conservation of volume 23
3 Numerical methods 25
3.1 Numerical integrators 25
3.2 Stiff problems 27
3.3 Runge Kutta methods 28
3.3.1 The class of Runge Kutta methods 28
3.3.2 Collocation methods. Gauss methods 30
vi CONTENTS
3.3.3 Existence and uniqueness of solutions in
implicit methods 33
3.4 Partitioned Runge Kutta methods 34
3.5 Runge Kutta Nystrom methods 36
3.6 Composition of methods. Adjoints 37
3.6.1 Composing methods 37
3.6.2 Adjoints 38
3.6.3 Finding Runge Kutta, Partitioned Runge
Kutta and Runge Kutta Nystrom adjoints 39
4 Order conditions 41
4.1 The order in Runge Kutta methods 41
4.2 The local error in Runge Kutta methods 43
4.3 The order in Partitioned Runge Kutta methods 45
4.4 The local error in Partitioned Runge Kutta methods 46
4.5 The order in Runge Kutta Nystrom methods 48
4.6 The local error in Runge Kutta Nystrom methods 50
5 Implementation 53
5.1 Variable step sizes 53
5.2 Embedded pairs 54
5.3 Numerical experience with variable step sizes 56
5.4 Implementing implicit methods 61
5.4.1 Reformulation of the equations 61
5.4.2 Solving the equations: functional iteration 62
5.4.3 Solving the equations: Newton like iteration 62
5.4.4 Starting the iterations 63
5.4.5 Stopping the iterations 64
5.4.6 The algebraic equations in the Runge Kutta
Nystrom case 64
5.5 Fourth order Gauss method 65
6 Symplectic integration 69
6.1 Symplectic methods 69
6.2 Symplectic Runge Kutta methods 72
6.3 Symplectic Partitioned Runge Kutta methods 76
6.4 Symplectic Runge Kutta Nystrom methods 77
6.5 Necessity of the symplecticness conditions 80
6.5.1 Preliminaries 80
6.5.2 Independence of the elementary differentials
in Hamiltonian problems 81
CONTENTS vii
6.5.3 Necessity of the symplecticness conditions in
the Partitioned Runge Kutta case 82
6.5.4 Other cases 85
7 Symplectic order conditions 87
7.1 Preliminaries 87
7.2 Order of symplectic Runge Kutta methods 88
7.3 Order of symplectic Partitioned methods 91
7.4 Order of symplectic Runge Kutta Nystrom methods 93
7.5 Homogeneous form of the order conditions 95
7.5.1 Motivation 95
7.5.2 The Partitioned Runge Kutta case 95
7.5.3 Other cases 97
8 Available symplectic methods 99
8.1 Symplecticness of the Gauss methods 99
8.2 Diagonally implicit Runge Kutta methods 100
8.2.1 General format 100
8.2.2 Specific methods 101
8.3 Other symplectic Runge Kutta methods 102
8.4 Explicit Partitioned Runge Kutta methods 103
8.4.1 General format 103
8.4.2 An alternative format 104
8.4.3 Specific methods: orders 1 and 2 105
8.4.4 Specific methods: order 3 107
8.4.5 Specific methods: order 4 out of order 3 108
8.4.6 Specific methods: order 4 109
8.4.7 Specific methods: concatenations of the
leap frog methods 110
8.5 Available symplectic Runge Kutta Nystrom methods 110
8.5.1 Implicit methods 110
8.5.2 Explicit methods: general format 110
8.5.3 Specific explicit methods 111
9 Numerical experiments 115
9.1 A comparison of symplectic integrators 115
9.1.1 Methods being compared 115
9.1.2 Results: Kepler s problem 117
9.1.3 Results: Henon Heiles problem 121
9.1.4 Results: computation of frequencies 123
9.2 Variable step sizes for symplectic methods 124
9.3 Conclusions and recommendations 127
viii CONTENTS
10 Properties of symplectic integrators 129
10.1 Backward error interpretation 129
10.1.1 An example 129
10.1.2 The general case 132
10.1.3 Application to variable step sizes 133
10.2 An alternative approach 134
10.3 Conservation of energy 136
10.3.1 Exact conservation: positive results 136
10.3.2 Exact conservation: negative results 137
10.3.3 Approximate conservation 139
10.4 KAM theory 141
11 Generating functions 143
11.1 The concept of generating function 143
11.1.1 Introduction 143
11.1.2 Generating functions of the first kind 143
11.1.3 Generating functions of the third kind 144
11.1.4 Generating functions of all kinds 145
11.2 Hamilton Jacobi equations 146
11.3 Integrators based on generating functions 147
11.4 Generating functions for Runge Kutta methods 149
11.5 Canonical order theory 150
11.5.1 General framework 150
11.5.2 The Partitioned Runge Kutta case 151
11.5.3 Elementary Hamiltonians 153
12 Lie formalism 155
12.1 The Poisson bracket 155
12.2 Lie operators and Lie series 156
12.2.1 Lie operators 156
12.2.2 The adjoint representation 157
12.2.3 Lie series 157
12.2.4 Multiplication of exponentials 159
12.3 The Baker Campbell Hausdorff formula 160
12.4 Application to fractional step methods 161
12.4.1 Introduction 161
12.4.2 The simplest splitting 163
12.4.3 Second order splitting 163
12.5 Extension to the non Hamiltonian case 164
13 High order methods 165
13.1 High order Lie methods 165
CONTENTS ix
13.1.1 Introduction 165
13.1.2 Yoshida s first approach: order 4 165
13.1.3 Yoshida s first approach: order 2r 167
13.1.4 Existence of symplectic methods of arbitrar¬
ily high orders 167
13.1.5 Yoshida s second approach 167
13.2 High order Runge Kutta Nystrom methods 170
13.2.1 Order 7 methods 170
13.2.2 Order 8 out of order 7 171
13.2.3 Connection with the Lie formalism 172
13.3 A comparison of order 8 symplectic integrators 173
13.3.1 Methods being compared 173
13.3.2 Results: Kepler s problem 174
13.3.3 Results: Henon Heiles problem 176
13.3.4 Results: computation of frequencies 176
13.3.5 Conclusions 177
14 Extensions 179
14.1 Partitioned Runge Kutta methods for nonseparable
Hamiltonian systems 179
14.2 Canonical B series 180
14.3 Conjugate symplectic methods. Trapezoidal rule 181
14.4 Constrained systems 182
14.5 General Poisson structures 184
14.6 Multistep methods 185
14.7 Partial differential equations 186
14.8 Reversible systems. Volume preserving flows 188
References 189
Symbol index 197
Index 201
|
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| id | DE-604.BV010023707 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:45:07Z |
| institution | BVB |
| isbn | 0412542900 9780486824109 |
| language | English |
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| physical | XII, 207 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
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| publisher | Chapman & Hall |
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| series | Applied mathematics and mathematical computation |
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| spelling | Sanz-Serna, J. M. Verfasser aut Numerical Hamiltonian problems J. M. Sanz-Serna and M. P. Calvo 1. ed. London u.a. Chapman & Hall 1994 XII, 207 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematics and mathematical computation 7 The purpose of this book is to present a unified introduction to the interdisciplinary and growing field of symplectic integration of Hamiltonian problems. These problems appear frequently in physics and other sciences and recent investigations suggest that they should be simulated by symplectic integrators, i.e. by numerical methods that preserve the symplectic structure of phase space, thus reproducing the main feature of Hamiltonian dynamics The authors explain in detail how to derive, analyse and use symplectic integrators. Virtually all currently available symplectic methods are reported using a coherent notation. Numerical Hamiltonian Problems includes five preliminary chapters on Hamiltonian problems and numerical methods, so that very little background knowledge should be required. Numerical examples are provided throughout Analyse numérique ram Globale analyse gtt PROBLEME HAMILTONIEN inriac Systèmes hamiltoniens ram méthode numérique inriac système hamiltonien inriac Hamiltonian systems Numerical analysis Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Numerische Integration (DE-588)4172168-8 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Numerische Integration (DE-588)4172168-8 s DE-188 Calvo, M. P. Verfasser aut Applied mathematics and mathematical computation 7 (DE-604)BV006188231 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006646084&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sanz-Serna, J. M. Calvo, M. P. Numerical Hamiltonian problems Applied mathematics and mathematical computation Analyse numérique ram Globale analyse gtt PROBLEME HAMILTONIEN inriac Systèmes hamiltoniens ram méthode numérique inriac système hamiltonien inriac Hamiltonian systems Numerical analysis Hamiltonsches System (DE-588)4139943-2 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Integration (DE-588)4172168-8 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4128130-5 (DE-588)4172168-8 |
| title | Numerical Hamiltonian problems |
| title_auth | Numerical Hamiltonian problems |
| title_exact_search | Numerical Hamiltonian problems |
| title_full | Numerical Hamiltonian problems J. M. Sanz-Serna and M. P. Calvo |
| title_fullStr | Numerical Hamiltonian problems J. M. Sanz-Serna and M. P. Calvo |
| title_full_unstemmed | Numerical Hamiltonian problems J. M. Sanz-Serna and M. P. Calvo |
| title_short | Numerical Hamiltonian problems |
| title_sort | numerical hamiltonian problems |
| topic | Analyse numérique ram Globale analyse gtt PROBLEME HAMILTONIEN inriac Systèmes hamiltoniens ram méthode numérique inriac système hamiltonien inriac Hamiltonian systems Numerical analysis Hamiltonsches System (DE-588)4139943-2 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Integration (DE-588)4172168-8 gnd |
| topic_facet | Analyse numérique Globale analyse PROBLEME HAMILTONIEN Systèmes hamiltoniens méthode numérique système hamiltonien Hamiltonian systems Numerical analysis Hamiltonsches System Numerisches Verfahren Numerische Integration |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006646084&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV006188231 |
| work_keys_str_mv | AT sanzsernajm numericalhamiltonianproblems AT calvomp numericalhamiltonianproblems |