CONTINUOUS SYMMETRIES, LIE ALGEBRAS, DIFFERENTIAL EQUATIONS AND COMPUTER ALGEBRA

by W-H Steeb

• ISBN13: 9789810228910
• ISBN10: 9810228910
• Format: Nonspecific Binding
• Copyright: 1996-09-30
• Publisher: World Scientific Publishing

Summary

Provides the standard techniques in the study of continuous symmetries and includes Painleve test and symmetries, invertible point transformation and symmetries, gauge theory and Yang-Mills theory and chaos. DLC: Differential equations.

Table of Contents

Introduction	1	(4)
Groups	5	(12)
Definitions and Examples	5	(10)
Computer Algebra Applications	15	(2)
Lie Groups and Lie Transformation Groups	17	(24)
Lie Groups	17	(2)
Lie Transformation Groups	19	(16)
Classical Groups	19	(9)
Affine Groups	28	(3)
Group Actions and Group Representations	31	(4)
General Remarks on Lie Groups	35	(5)
Computer Algebra Applications	40	(1)
Infinitesimal Transformations	41	(16)
Introduction	41	(4)
Infinitesimal Generators	45	(5)
Multi-Parameter Lie Groups of Transformations	50	(5)
Computer Algebra Applications	55	(2)
Lie Algebras	57	(26)
Definition and Examples	57	(3)
Lie Algebras and Bose and Fermi Operators	60	(1)
Lie Algebras and Lie Groups	61	(5)
Classical Lie Algebras	66	(4)
Important Concepts and Examples	70	(9)
Computer Algebra Applications	79	(4)
Introductory Examples	83	(18)
The One-Dimensional Linear Wave Equation	83	(4)
The One-Dimensional Diffusion Equation	87	(2)
Stationary Flow	89	(3)
Gauge Transformation	92	(7)
Computer Algebra Applications	99	(2)
Differential Forms and Tensor Fields	101	(22)
Vector Fields and Tangent Bundles	101	(2)
Differential Forms and Contangent Bundles	103	(4)
The Exterior Derivative	107	(2)
Pull-Back Maps	109	(1)
The Interior Product or Contraction	110	(1)
Riemannian Manifolds	111	(3)
Hodge Star Operator	114	(3)
Computer Algebra Applications	117	(6)
Lie Derivative and Invariance	123	(14)
Introduction	123	(1)
Definitions	123	(4)
Invariance and Conformal Invariance	127	(9)
Computer Algebra Applications	136	(1)
Invariance of Differential Equations	137	(40)
Prolongation of Vector Fields	137	(12)
Introductory Example	137	(3)
Local Jet Bundle Formalism	140	(5)
Prolongation of Vector Fields	145	(2)
Partial Differential Equations on Jet Bundles	147	(2)
Invariance of Differential Equations	149	(9)
Similarity Solutions	158	(9)
Transforming Solutions to Solutions	167	(2)
Direct Method	169	(2)
Maps and Invariants	171	(4)
Computer Algebra Applications	175	(2)
Lie-Backlund Vector Fields	177	(14)
Definitions and Examples	177	(9)
Invariant Solutions	186	(3)
Computer Algebra Applications	189	(2)
Differential Equation for a Given Lie Algebra	191	(8)
Lie Point Symmetry Vector Fields	191	(5)
Lie-Backlund Vector Fields	196	(2)
Computer Algebra Applications	198	(1)
A List of Lie Symmetry Vector Fields	199	(20)
Recursion Operators	219	(14)
Gateaux Derivative	219	(8)
Definition and Examples	227	(5)
Computer Algebra Applications	232	(1)
Backlund Transformations	233	(8)
Definitions	233	(3)
Examples	236	(4)
Computer Algebra Applications	240	(1)
Lax Representations	241	(8)
Definitions	241	(2)
Examples	243	(5)
Computer Algebra Applications	248	(1)
Conservation Laws	249	(26)
Basic Concepts	249	(6)
Exterior Differential Systems	255	(6)
Cartan Fundamental Form	261	(12)
Computer Algebra Applications	273	(2)
Symmetries and Painleve Test	275	(26)
Introduction	275	(1)
Ordinary Differential Equations	276	(5)
Invertible Point Transformation	281	(4)
Partial Differential Equations	285	(3)
Symmetries by Truncated Expansions	288	(3)
Painleve Test and Recursion Operators	291	(2)
Singular Manifold and Similarity Variables	293	(6)
Integrability Concepts	299	(1)
Computer Algebra Applications	300	(1)
Lie Algebra Valued Differential Forms	301	(20)
Introduction	301	(3)
Covariant Exterior Derivative	304	(2)
Yang-Mills and Self-Dual Yang-Mills Equations	306	(7)
Yang-Mills Equation and Chaos	313	(2)
Soliton Equations	315	(4)
Computer Algebra Applications	319	(2)
Bose Operators and Lie Algebras	321	(10)
Embedding and Bose Operators	321	(3)
Examples	324	(2)
Embedding and Bose Field Operators	326	(4)
Computer Algebra Applications	330	(1)
Computer Algebra	331	(10)
Computer Algebra Packages	331	(1)
Programs for Lie Symmetries	332	(4)
Basic REDUCE Commands	336	(2)
Examples	338	(3)
A Differentiable Manifolds	341	(8)
Bibliography	349	(8)
Index	357