Gespeichert in:
| Titel: | Nonlinear mechanics, groups and symmetry |
|---|---|
| Von: |
by Yu. A. Mitropolsky and A. K. Lopatin
|
| Person: |
Mitropolʹskij, Jurij Alekseevič
1917-2008 Verfasser aut Lopatin, Aleksej K. |
| Hauptverfassende: | , |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Dordrecht u.a.
Kluwer
1995
|
| Schriftenreihe: | Mathematics and its applications
319 |
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006837272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | IX, 377 S. |
| ISBN: | 079233339X |
Internformat
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| 240 | 1 | 0 | |a Teoretiko-gruppovoj podchod v asimptotičeskich metodach nelinejnoj mechaniki |
| 245 | 1 | 0 | |a Nonlinear mechanics, groups and symmetry |c by Yu. A. Mitropolsky and A. K. Lopatin |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer |c 1995 | |
| 300 | |a IX, 377 S. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 319 | |
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Datensatz im Suchindex
| _version_ | 1819323893245018112 |
|---|---|
| adam_text | Contents
Introduction 1
1 Vector Fields, Algebras and Groups Generated by a System of
Ordinary Differential Equations and their Properties 13
1.1. The system of differential equations and its generating Lie algebra . 13
1.1.1. Basic definitions 13
1.1.2. Examples 17
1.2. Lie series as the solution of the system of ordinary differential equa¬
tions and its properties. The generating group of the system 18
1.2.1. Basic properties of Lie series 18
1.2.2. Pointwise transformations 25
1.2.3. On a connection between generating Lie algebras of a system
and certain groups 28
1.3. The change of variables in a differential system. Carnpbell Hausdorff
formula 35
1.3.1. The technique for variable change 35
1.3.2. Variable changes using Lie series. The Lie transformation . . 37
1.4. Lie theory of systems of ordinary differential equations admitting a
group of transformations 38
1.4.1. Basic definitions and the theorem on invariance 38
1.4.2. What does the knowledge of the algebra of symmetry of a
system give for its integration? 40
1.4.3. Lie s result on the reduction of an arbitrary algebra of the
symmetry of a system to finite algebra 42
1.4.4. Examples 44
1.4.5. Connection of above mentioned results on invariance with
Lie s theory of extended operators 47
2 Decomposition of Systems of Ordinary Differential Equations 51
2.1. Algebraic reducibility of systems of linear ordinary differential equa¬
tions with variable coefficients 51
2.1.1. Principal definitions and theorems 51
2.1.2. Basic and auxiliary statements 54
2.1.3. The main algorithm. Examples 67
V
vi CONTENTS
2.2. A generalized linear method for the decomposition of a differential
system to block triangular form 70
2.2.1. Formulation of the problem 70
2.2.2. The Danilevsky s method 71
2.2.3. Method of decomposition by a nilpotent component 76
2.2.4. The generalized linear method 83
2.3. The algebraic reductibility of systems of linear differential equations
with variable coefficients, the matrices of which c.omrnutate with
their integral 85
2.4. Decomposition of systems of nonlinear differential equations 88
2.4.1. Formulation of the problem 88
2.4.2. Some auxiliary statements 90
2.4.3. The principal theorems 94
2.4.4. Examples 99
2.5. Reduction of the number of variables in the system of ordinary dif¬
ferential equations 101
2.5.1. Formulation of the problem 101
2.5.2. The principal theorems 102
2.5.3. Dynamical motions of a satellite 103
2.6. Algebraically reducible systems 106
2.6.1. Formulation of the problem and principal theorems 106
2.6.2. A group theoretical view to the results 107
2.6.3. Certain mechanical models 109
3 Asymptotic decomposition of systems of ordinary differential
equations with a small parameter 115
3.1. The general scheme of the asymptotic decomposition algorithm . . .115
3.1.1. The main algorithm 115
3.1.2. The centralized system and its group theoretical properties . 119
3.1.3. Two approaches in the construction of the centralized system 120
3.2. Basic theorems on the integration of the centralized system 121
3.2.1. Results related to the separation of motions 121
3.2.2. Results related to the separation of variables into fast and slow 125
3.3. Reduction of operator equations to differential equations 129
3.3.1. Reduction of operator equations to the Jacobian systems . . 129
3.3.2. Various approaches to solving the Jacobian systems arrived
at via various methods of nonlinear mechanics 131
3.4. Realization of an asymptotic decomposition algorithm in the domain
of the existing first integrals of a system of zero approximation . . . 133
3.4.1. Formulation of the problem 133
3.4.2. Construction of the domain Ho(x) 133
3.4.3. Investigation of the asymptotic decomposition algorithm . . . 135
3.5. Substantiation of the asymptotic decomposition algorithm for a fi¬
nite number of approximations 143
3.5.1. Shortened transformations and the centralized system of m
approximations 143
CONTENTS vii
3.5.2. Justification of the asymptotic decomposition algorithm bas¬
ing on Poincare s theorem 145
3.6. The asymptotic method of separation of variables by Krylov, Bo
goliubov and Mitropolsky (KBM method) and the asymptotic de¬
composition method 151
3.6.1. General remarks 151
3.6.2. Systems of the standard form. The Bogoliubov projector . . 152
3.6.3. Systems of nonlinear mechanics with several fast variables . . 156
4 Asymptotic Decomposition of Almost Linear Systems of Differ¬
ential Equations with Constant Coefficients and Perturbations in
the Form of Polynomials 159
4.1. The generating algebras B and 3 of the initial system 159
4.1.1. Passing to the matrix of a simple structure 159
4.1.2. General settings 160
4.1.3. The structure of the operators of the system of zero approx¬
imation 161
4.2. Reduction of the solution of operator equations to a solution of a
system of algebraic equations 163
4.2.1. Basic equations 163
4.2.2. Three approaches to the solution of the basic equations . . . 164
4.3. Construction of a centralized system and finding the reducing trans¬
formations 165
4.3.1. The basic algorithm 165
4.3.2. The existence conditions of nonzero projections 167
4.4. The structure of a centralized system. The basic theorems on de¬
composition and separation of motions 168
4.4.1. The basic theorem 168
4.4.2. Separation of variables in the centralized system 169
4.4.3. Some other results 175
4.4.4. A sufficient criterion of decomposability 177
4.5. Models based on Lotka Volterra system 181
4.6. Models based on the Van der Pol system 194
4.7. A model of the point motion on a sphere 208
4.8. The asymptotic decomposition and the normal form methods .... 215
4.8.1. The normal form method 215
4.8.2. Comparison of normal form and asymptotic decomposition
methods 216
5 Asymptotic Decomposition of Differential Systems with Small Pa¬
rameter in the Representation Space of Finite dimensional Lie
Group 219
5.1. Formulation of the problem 219
5.2. An asymptotic decomposition algorithm in a representation space of
a finite dimensional Lie group 222
viii CONTENTS
5.2.1. Group theoretical properties of the system of zero approxi¬
mation 222
5.2.2. The main algorithm. Reduction of operator equations to
systems of linear algebraic equations 223
5.2.3. Some results on the separation of variables in a centralized
system 228
5.2.4. Advantages of passing from general linear group GL{n) to
its subgroup G{Bh) 229
5.3. Models connected with 50(2) 230
5.4. The motion of a point on a sphere (a model of the motion connected
with 50(3)) 234
5.5. Dynamical maneuvers of a satellite (perturbed motion on 5 0(2) x
50(3) x 5O(3) x 50(3)) 240
5.6. Almost invariant systems of differential equations with a compact
Lie group of invariance 248
5.6.1. Formulation of the problem 248
5.6.2. The main algorithm 250
5.6.3. Examples 255
6 Asymptotic Decomposition of Differential Systems where Zero
Approximation has Special Properties 259
6.1. .Systems of zero approximation having a known symmetry algebra . . 259
6.1.1. Formulation of the problem 259
6.1.2. Basic theorems 261
6.1.3. Example 264
6.2. Asymptotic decomposition algorithm when zero approximation is
decomposable 267
6.2.1. Basic assumptions about the system of zero approximation . 267
6.2.2. Conditions of decomposability for a perturbed system . . . .268
6.3. Linear systems with constant coefficients and a small parameter . . . 276
6.3.1. Algorithm realization 276
6.3.2. The centralized system is always decomposable 277
6.3.3. The case of decomposability of the zero approximation .... 281
6.3.4. A model of a mechanical system 286
6.4. The general case of the structure of a matrix of zero approximation
system 295
6.4.1. Formulation of the problem and realization of the asymptotic
decomposition algorithm 295
6.4.2. Basic theorems 298
6.5. Method of local asymptotic decomposition 299
6.5.1. Dynamics of flying apparatus 300
6.5.2. A model of gas dynamics 303
CONTENTS ix
7 Asymptotic Decomposition of Pfaffian Systems with a Small Pa¬
rameter 305
7.1. Reduction of the general problem of integration of a partial differ¬
ential equation system to that of a Pfaffian system 305
7.1.1. The equivalence of the integration problem for a partial dif¬
ferential equation system and that for a Pfaffian system . . . 305
7.1.2. The perturbation problem for a Pfaffian system 309
7.2. Asymptotic decomposition of completely integrable Pfaffian systems
with a small parameter 310
7.2.1. Formulation of the problem and the main algorithm 310
7.2.2. Theorems on the integration of the centralized system . . . .313
7.2.3. Justification of the asymptotic decomposition algorithm . . .316
7.2.4. Not completely integrable systems 319
7.3. Searching for integrals of a Pfaffian system in involution 320
7.3.1. The chain of integral elements 320
7.3.2. Theorems on integration of the centralized system 326
7.4. Asymptotic decomposition of a Pfaffian system, which is in involu¬
tion, with a small parameter 329
7.4.1. The reduction of the general problem to the perturbation
problem for completely integrable systems 329
7.4.2. Model of a Pfaffian system for a wave equation 331
Appendix 339
A: Lie series and Lie transformation 339
B: The direct product of matrices 342
Bl: Definition 342
B2: Systems of matrix equations 342
C: Conditions for the solvability of systems of linear equations 342
D: Elements of Lie group analysis of differential equations on the basis of
the theory of extended operators 345
Dl: One parameter group and its infinitesimal operator 345
D2. Theory of extension 346
Bibliographical Comments 349
References 353
Index 373
|
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| author | Mitropolʹskij, Jurij Alekseevič 1917-2008 Lopatin, Aleksej K. |
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| id | DE-604.BV010276567 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T09:51:11Z |
| institution | BVB |
| isbn | 079233339X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006837272 |
| oclc_num | 246725888 |
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| physical | IX, 377 S. |
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| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spellingShingle | Mitropolʹskij, Jurij Alekseevič 1917-2008 Lopatin, Aleksej K. Nonlinear mechanics, groups and symmetry Mathematics and its applications Nichtlineare Mechanik (DE-588)4042095-4 gnd Asymptotische Methode (DE-588)4287476-2 gnd Störungstheorie (DE-588)4128420-3 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| subject_GND | (DE-588)4042095-4 (DE-588)4287476-2 (DE-588)4128420-3 (DE-588)4072157-7 |
| title | Nonlinear mechanics, groups and symmetry |
| title_alt | Teoretiko-gruppovoj podchod v asimptotičeskich metodach nelinejnoj mechaniki |
| title_auth | Nonlinear mechanics, groups and symmetry |
| title_exact_search | Nonlinear mechanics, groups and symmetry |
| title_full | Nonlinear mechanics, groups and symmetry by Yu. A. Mitropolsky and A. K. Lopatin |
| title_fullStr | Nonlinear mechanics, groups and symmetry by Yu. A. Mitropolsky and A. K. Lopatin |
| title_full_unstemmed | Nonlinear mechanics, groups and symmetry by Yu. A. Mitropolsky and A. K. Lopatin |
| title_short | Nonlinear mechanics, groups and symmetry |
| title_sort | nonlinear mechanics groups and symmetry |
| topic | Nichtlineare Mechanik (DE-588)4042095-4 gnd Asymptotische Methode (DE-588)4287476-2 gnd Störungstheorie (DE-588)4128420-3 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| topic_facet | Nichtlineare Mechanik Asymptotische Methode Störungstheorie Gruppentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006837272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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