Geometrical Theory of Dynamical Systems and Fluid Flows

by Kambe, Tsutomu

• ISBN13: 9789812388063
• ISBN10: 9812388060
• Format: Hardcover
• Copyright: 2004-11-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

Preface	v
I. Mathematical Bases
1. Manifolds, Flows, Lie Groups and Lie Algebras	3	(42)
1.1 Dynamical Systems	3	(1)
1.2 Manifolds and Diffeomorphisms	4	(4)
1.3 Flows and Vector Fields	8	(5)
1.3.1 A steady flow and its velocity field	8	(2)
1.3.2 Tangent vector and differential operator	10	(1)
1.3.3 Tangent space	11	(1)
1.3.4 Time-dependent (unsteady) velocity field	12	(1)
1.4 Dynamical Trajectory	13	(5)
1.4.1 Fiber bundle (tangent bundle)	13	(1)
1.4.2 Lagrangian and Hamiltonian	14	(2)
1.4.3 Legendre transformation	16	(2)
1.5 Differential and Inner Product	18	(3)
1.5.1 Covector (1-form)	18	(2)
1.5.2 Inner (scalar) product	20	(1)
1.6 Mapping of Vectors and Covectors	21	(4)
1.6.1 Push-forward transformation	21	(2)
1.6.2 Pull-back transformation	23	(1)
1.6.3 Coordinate transformation	24	(1)
1.7 Lie Group and Invariant Vector Fields	25	(2)
1.8 Lie Algebra and Lie Derivative	27	(8)
1.8.1 Lie algebra, adjoint operator and Lie bracket	27	(2)
1.8.2 An example of the rotation group SO(3)	29	(1)
1.8.3 Lie derivative and Lagrange derivative	30	(5)
1.9 Diffeomorphisms of a Circle S¹	35	(1)
1.10 Transformation of Tensors and Invariance	36	(9)
1.10.1 Transformations of vectors and metric tensors	36	(2)
1.10.2 Covariant tensors	38	(1)
1.10.3 Mixed tensors	39	(3)
1.10.4 Contravariant tensors	42	(3)
2. Geometry of Surfaces in R³	45	(32)
2.1 First Fundamental Form	45	(5)
2.2 Second Fundamental Form	50	(2)
2.3 Gauss's Surface Equation and an Induced Connection	52	(2)
2.4 Gauss-Mainardi-Codazzi Equation and Integrability	54	(2)
2.5 Gaussian Curvature of a Surface	56	(7)
2.5.1 Riemann tensors	56	(2)
2.5.2 Gaussian curvature	58	(1)
2.5.3 Geodesic curvature and normal curvature	59	(1)
2.5.4 Principal curvatures	60	(3)
2.6 Geodesic Equation	63	(1)
2.7 Structure Equations in Differential Forms	64	(5)
2.7.1 Smooth surfaces in R³ and integrability	64	(2)
2.7.2 Structure equations	66	(2)
2.7.3 Geodesic equation	68	(1)
2.8 Gauss Spherical Map	69	(1)
2.9 Gauss-Bonnet Theorem I	70	(2)
2.10 Gauss Bonnet Theorem II	72	(2)
2.11 Uniqueness: First and Second Fundamental Tensors	74	(3)
3. Riemannian Geometry	77	(50)
3.1 Tangent Space	77	(3)
3.1.1 Tangent vectors and inner product	77	(1)
3.1.2 Riemannian metric	78	(1)
3.1.3 Examples of metric tensor	79	(1)
3.2 Covariant Derivative (Connection)	80	(2)
3.2.1 Definition	80	(1)
3.2.2 Time-dependent case	81	(1)
3.3 Riemannian Connection	82	(1)
3.3.1 Definition	82	(1)
3.3.2 Christoffel symbol	83	(1)
3.4 Covariant Derivative along a Curve	83	(2)
3.4.1 Derivative along a parameterized curve	83	(1)
3.4.2 Parallel translation	84	(1)
3.4.3 Dynamical system of an invariant metric	84	(1)
3.5 Structure Equations	85	(6)
3.5.1 Structure equations and connection forms	85	(3)
3.5.2 Two-dimensional surface M²	88	(1)
3.5.3 Example: Poincaré surface (I)	89	(2)
3.6 Geodesic Equation	91	(5)
3.6.1 Local coordinate representation	91	(1)
3.6.2 Group-theoretic representation	92	(1)
3.6.3 Example: Poincaré surface (II)	93	(3)
3.7 Covariant Derivative and Parallel Translation	96	(4)
3.7.1 Parallel translation again	96	(2)
3.7.2 Covariant derivative again	98	(1)
3.7.3 A formula of covariant derivative	98	(2)
3.8 Arc-Length	100	(2)
3.9 Curvature Tensor and Curvature Transformation	102	(6)
3.9.1 Curvature transformation	102	(1)
3.9.2 Curvature tensor	103	(2)
3.9.3 Sectional curvature	105	(2)
3.9.4 Ricci tensor and scalar curvature	107	(1)
3.10 Jacobi Equation	108	(6)
3.10.1 Derivation	108	(3)
3.10.2 Initial behavior of Jacobi field	111	(1)
3.10.3 Time-dependent problem	112	(1)
3.10.4 Two-dimensional problem	113	(1)
3.10.5 Isotropic space	114	(1)
3.11 Differentiation of Tensors	114	(3)
3.11.1 Lie derivatives of 1-form and metric tensor	114	(1)
3.11.2 Riemannian connection	115	(1)
3.11.3 Covariant derivative of tensors	116	(1)
3.12 Killing Fields	117	(4)
3.12.1 Killing vector field X	117	(1)
3.12.2 Isometry	118	(1)
3.12.3 Positive curvature and simplified Jacobi equation	118	(1)
3.12.4 Conservation of (X,T) along λ(s)	119	(1)
3.12.5 Killing tensor field	120	(1)
3.13 Induced Connection and Second Fundamental Form	121	(6)
II. Dynamical Systems
4. Free Rotation of a Rigid Body	127	(26)
4.1 Physical Background	127	(4)
4.1.1 Free rotation and Euler's top	127	(3)
4.1.2 Integrals of motion	130	(1)
4.1.3 Lie-Poisson bracket and Hamilton's equation	131	(1)
4.2 Transformations (Rotations) by SO(3)	131	(4)
4.2.1 Transformation of reference frames	132	(1)
4.2.2 Right-invariance and left-invariance	133	(2)
4.3 Commutator and Riemannian Metric	135	(2)
4.4 Geodesic Equation	137	(2)
4.4.1 Left-invariant dynamics	137	(1)
4.4.2 Right-invariant dynamics	138	(1)
4.5 Bi-Invariant Riemannian Metrices	139	(4)
4.5.1 SO(3) is compact	140	(1)
4.5.2 Ad-invariance and bi-invariant metrices	140	(2)
4.5.3 Connection and curvature tensor	142	(1)
4.6 Rotating Top as a Bi-Invariant System	143	(10)
4.6.1 A spherical top (euclidean metric)	143	(1)
4.6.2 An asymmetrical top (Riemannian metric)	144	(2)
4.6.3 Symmetrical top and its stability	146	(3)
4.6.4 Stability and instability of an asymmetrical top	149	(1)
4.6.5 Supplementary notes to §4.6.3	150	(3)
5. Water Waves and KdV Equation	153	(18)
5.1 Physical Background: Long Waves in Shallow Water	154	(4)
5.2 Simple Diffeomorphic Flow	158	(3)
5.2.1 Commutator and metric of D(S¹)	158	(2)
5.2.2 Geodesic equation on D(S¹)	160	(1)
5.2.3 Sectional curvatures on D(S¹)	160	(1)
5.3 Central Extension of D(S¹)	161	(1)
5.4 KdV Equation as a Geodesic Equation on D(S¹)	161	(2)
5.5 Killing Field of KdV Equation	163	(5)
5.5.1 Killing equation	163	(1)
5.5.2 Isometry group	163	(1)
5.5.3 Integral invariant	164	(1)
5.5.4 Sectional curvature	165	(1)
5.5.5 Conjugate point	166	(2)
5.6 Sectional Curvatures of KdV System	168	(3)
6 Hamiltonian Systems: Chaos, Integrability and Phase Transition	171	(20)
6.1 A Dynamical System with Self-Interaction	171	(4)
6.1.1 Hamiltonian and metric tensor	171	(2)
6.1.2 Geodesic equation	173	(1)
6.1.3 Jacobi equation	173	(1)
6.1.4 Metric and covariant derivative	174	(1)
6.2 Two Degrees of Freedom	175	(2)
6.2.1 Potentials	175	(1)
6.2.2 Sectional curvature	176	(1)
6.3 Hénon-Heiles Model and Chaos	177	(1)
6.3.1 Conventional method	177	(1)
6.3.2 Evidence of chaos in a geometrical aspect	177	(1)
6.4 Geometry and Chaos	178	(3)
6.5 Invariants in a Generalized Model	181	(2)
6.5.1 Killing vector field	181	(2)
6.5.2 Another integrable case	183	(1)
6.6 Topological Signature of Phase Transitions	183	(8)
6.6.1 Morse function and Euler index	184	(1)
6.6.2 Signatures of phase transition	185	(1)
6.6.3 Topological change in the mean-field XY model	186	(5)
III. Flows of Ideal Fluids
7 Gauge Principle and Variational Formulation	191	(62)
7.1 Introduction: Fluid Flows and Field Theory	191	(2)
7.2 Lagrangians and Variational Principle	193	(5)
7.2.1 Galilei-invariant Lagrangian	193	(3)
7.2.2 Hamilton's variational formulations	196	(1)
7.2.3 Lagrange's equation	197	(1)
7.3 Conceptual Scenario of the Gauge Principle	198	(4)
7.4 Global Gauge Transformation	202	(1)
7.5 Local Gauge Transformation	202	(3)
7.5.1 Covariant derivative	203	(1)
7.5.2 Lagrangian	204	(1)
7.6 Symmetries of Flow Fields	205	(3)
7.6.1 Translational transformation	206	(1)
7.6.2 Rotational transformation	206	(1)
7.6.3 Relative displacement	207	(1)
7.7 Laws of Translational Transformation	208	(4)
7.7.1 Local Galilei transformation	208	(1)
7.7.2 Determination of gauge field Α	209	(1)
7.7.3 Irrotational fields ξ(χ) and u(χ)	210	(2)
7.8 Fluid Flows as Material Motion	212	(4)
7.8.1 Lagrangian particle representation	212	(3)
7.8.2 Lagrange derivative and Lie derivative	215	(1)
7.8.3 Kinematical constraint	216	(1)
7.9 Gauge-Field Lagrangian LA (Translational Symmetry)	216	(2)
7.9.1 A possible form	216	(1)
7.9.2 Lagrangian of background thermodynamic state	217	(1)
7.10 Hamilton's Principle for Potential Flows	218	(6)
7.10.1 Lagrangian	218	(1)
7.10.2 Material variations: irrotational and isentronic	219	(1)
7.10.3 Constraints for variations	220	(1)
7.10.4 Action principle for Lp	221	(3)
7.11 Rotational Transformations	224	(2)
7.11.1 Orthogonal transformation of velocity	224	(1)
7.11.2 Infinitesimal transformations	225	(1)
7.12 Gauge Transformation (Rotation)	226	(3)
7.12.1 Local gauge transformation	226	(1)
7.12.2 Covariant derivative	227	(1)
7.12.3 Gauge principle	228	(1)
7.12.4 Transformation law of the gauge field Ω	228	(1)
7.13 Gauge-Field Lagrangian LB (Rotational Symmetry)	229	(2)
7.14 Riot Savart's Law	231	(2)
7.14.1 Vector potential of mass flux	231	(2)
7.14.2 Vorticity as a gauge field	233	(1)
7.15 Hamilton's Principle for an Ideal Fluid (Rotational Flows)	233	(9)
7.15.1 Constitutive conditions	234	(1)
7.15.2 Lagrangian and its variations	234	(1)
7.15.3 Material variation: rotational and isentropic	235	(2)
7.15.4 Euler's equation of motion	237	(1)
7.15.5 Conservations of momentum and energy	238	(2)
7.15.6 Noether's theorem for rotations	240	(2)
7.16 Local Symmetries in α-Space	242	(8)
7.16.1 Equation of motion in α-space	242	(2)
7.16.2 Vorticity equation and local rotation symmetry	244	(2)
7.16.3 Vorticity equation in the χ-space	246	(2)
7.16.4 Kelvin's circulation theorem	248	(1)
7.16.5 Lagrangian of the gauge field	249	(1)
7.17 Conclusions	250	(3)
8 Volume-Preserving Flows of an Ideal Fluid	253	(40)
8.1 Fundamental Concepts	254	(6)
8.1.1 Volume-preserving diffeomorphisms	254	(3)
8.1.2 Right-invariant fields	257	(3)
8.2 Basic Tools	260	(4)
8.2.1 Commutator	260	(1)
8.2.2 Divergence-free connection	261	(1)
8.2.3 Coadjoint action ad*	262	(1)
8.2.4 Formulas in R³ space	263	(1)
8.3 Geodesic Equation	264	(2)
8.4 Jacobi Equation and Frozen Field	266	(2)
8.5 Interpretation of Riemannian Curvature of Fluid Flows	268	(7)
8.5.1 Flat connection	268	(1)
8.5.2 Pressure gradient as an agent yielding curvature	269	(2)
8.5.3 Instability in Lagrangian particle sense	271	(2)
8.5.4 Time evolution of Jacobi field	273	(1)
8.5.5 Stretching of line-elements	273	(2)
8.6 Flows on a Cubic Space (Fourier Representation)	275	(2)
8.7 Lagrangian Instability of Parallel Shear Flows	277	(8)
8.7.1 Negative sectional curvatures	277	(2)
8.7.2 Stability of a plane Couette flow	279	(5)
8.7.3 Other parallel shear flows	284	(1)
8.8 Steady Flows and Beltrami Flows	285	(5)
8.8.1 Steady flows	285	(2)
8.8.2 A Beltrami flow	287	(2)
8.8.3 ABC flow	289	(1)
8.9 Theorem: α¹B = -iudα¹w + df	290	(3)
9. Motion of Vortex Filaments	293
9.1 A Vortex Filament	294	(3)
9.2 Filament Equation	297	(4)
9.3 Basic Properties	301	(3)
9.3.1 Left-invariance and right-invariance	301	(1)
9.3.2 Landau-Lifshitz equation	302	(1)
9.3.3 Lie Poisson bracket and Hamilton's equation	302	(2)
9.3.4 Metric and loop algebra	304	(1)
9.4 Geometrical Formulation and Geodesic Equation	304	(2)
9.5 Vortex Filaments as a Bi-Invariant System	306	(4)
9.5.1 Circular vortex filaments	306	(2)
9.5.2 General vortex filaments	308	(1)
9.5.3 Integral invariants	308	(2)
9.6 Killing Fields on Vortex Filaments	310	(4)
9.6.1 A rectilinear vortex	310	(1)
9.6.2 A circular vortex	311	(2)
9.6.3 A helical vortex	313	(1)
9.7 Sectional Curvature and Geodesic Stability	314	(1)
9.7.1 Killing fields	314	(1)
9.7.2 General tangent field X	315	(1)
9.8 Central Extension of the Algebra of Filament Motion	315	(6)
IV. Geometry of Integrable Systems
10. Geometric Interpretations of Sine-Gordon Equation	321	(30)
10.1 Pseudosphere: A Geometric Derivation of SG	321	(3)
10.2 Bianchi-Lie Transformation	324	(2)
10.3 B4cklund Transformation of SG Equation	326	(3)
11. Integrable Surfaces: Riemannian Geometry and Group Theory	329	(1)
11.1 Basic Ideas	329	(1)
11.2 Iseudospherical Surfaces: SG, KdV, mKdV, ShG	330	(3)
11.3 Spherical Surfaces: NLS, SG, NSM	333	(7)
11.3.1 Nonlinear Schrodinger equation	333	(2)
11.3.2 Sine-Gordon equation revisited	335	(1)
11.3.3 Nonlinear sigma model and SG equation	336	(2)
11.3.4 Spherical and pseudospherical surfaces	338	(2)
11.4 Backlund Transformations Revisited	340	(3)
11.4.1 A Bäcklund transformation	340	(2)
11.4.2 Self-Backlund transformation	342	(1)
11.5 Immersion of Integrable Surfaces on Lie Groups	343	(5)
11.5.1 A surface Σ² in R³	343	(1)
11.5.2 Surfaces on Lie groups and Lie algebras	344	(2)
11.5.3 Nonlinear Schrödinger surfaces	346	(2)
11.6 Mapping of Integrable Systems to Spherical Surfaces	348	(3)
Appendix A Topological Space and Mappings	351	(1)
A.1 Topology	351	(1)
A.2 Mappings	351	(2)
Appendix B Exterior Forms, Products and Differentials	353	(1)
B.1 Exterior Forms	353	(1)
B.2 Exterior Products (Multiplications)	355	(1)
B.3 Exterior Differentiations	357	(1)
B.4 Interior Products and Caftan's Formula	358	(1)
B.5 Vector Analysis in R³	358	(1)
B.6 Volume Form and Its Lie Derivative	361	(1)
B.7 Integration of Forms	362	(3)
B.7.1 Stokes's theorem	362	(1)
B.7.2 Integral and pull-back	363	(2)
Appendix C Lie Groups and Rotation Groups	365	(1)
C.1 Various Lie Groups	365	(1)
C.2 One-Parameter Subgroup and Lie Algebra	366	(1)
C.3 Rotation Group SO(η)	367	(1)
C.4 so(3)	368	(3)
Appendix D A Curve and a Surface in R³	371	(1)
D.1 Frenet-Serret Formulas for a Space Curve	371	(1)
D.2 A Plane Curve in R² and Gauss Map	372	(1)
D.3 A Surface Defined by z = f(x, y) in R³	373	(2)
Appendix E Curvature Transformation	375	(4)
Appendix F Function Spaces Lp, Hs and Orthogonal Decomposition	379	(2)
Appendix G Derivation of KdV Equation for a Shallow Water Wave	381	(1)
G.1 Basic Equations and Boundary Conditions	381	(1)
G.2 Long Waves in Shallow Water	382	(3)
Appendix H Two-Cocycle, Central Extension and Bott Cocycle	385	(1)
H.1 Two-Cocycle and Central Extension	385	(1)
H.2 Bott Cocycle	387	(1)
H.3 Gelfand-Fuchs Cocycle: An Extended Algebra	388	(3)
Appendix I Additional Comment on the Gauge Theory of §7.3	391	(2)
Appendix J Frobenius Integration Theorem and Pfaffian System	393	(2)
Appendix K Orthogonal Coordinate Net and Lines of Curvature	395	(4)
References	399	(8)
Index	407