Geometrical Theory of Dynamical Systems and Fluid Flows

The following sections are included:

• Preface to Revised Edition

• Preface to First Edition

• Contents

In geometrical theory of dynamical systems, fundamental notions and tools are manifolds, diffeomorphisms, flows, exterior algebras and Lie algebras.

Most concepts in differential geometry spring from the geometry of surfaces in ℝ³.

We consider the “inner” geometry of a manifold which is not a part of an euclidean space. We consider only tangential vectors, and any vector normal to the manifold is not available. We presuppose that each tangent bundle possesses an inner product depending on points of its base space smoothly. The space is curved in general. The Riemannian curvature of a manifold governs the behavior of geodesics on it and corresponding dynamical system. Dimension of the manifold is not always finite.

The following sections are included:

• Physical Background
  • Free rotation and Euler's top
  • Integrals of motion
  • Lie–Poisson bracket and Hamilton's equation

• Transformations (Rotations) by SO(3)
  • Transformation of reference frames
  • Right-invariance and left-invariance

• Commutator and Riemannian Metric

• Geodesic Equation
  • Left-invariant dynamics
  • Right-invariant dynamics

• Bi-Invariant Riemannian Metrices
  • SO(3) is compact
  • Ad-invariance and bi-invariant metrices
  • Connection and curvature tensor

• Rotating Top as a Bi-Invariant System
  • A spherical top (euclidean metric)
  • An asymmetrical top (Riemannian metric)
  • Symmetrical top and its stability
  • Stability and instability of an asymmetrical top
  • Supplementary notes to §4.6.3

The following sections are included:

• Physical Background: Long Waves in Shallow Water

• Simple Diffeomorphic Flow
  • Commutator and metric of D(S¹)
  • Geodesic equation on D(S¹)
  • Sectional curvatures on D(S¹)

• Central Extension of D(S¹)

• KdV Equation as a Geodesic Equation on

• Killing Field of KdV Equation
  • Killing equation
  • Isometry group
  • Integral invariant
  • Sectional curvature
  • Conjugate point

• Sectional Curvatures of KdV System

The following sections are included:

• A Dynamical System with Self-Interaction
  • Hamiltonian and metric tensor
  • Geodesic equation
  • Jacobi equation
  • Metric and covariant derivative

• Two Degrees of Freedom
  • Potentials
  • Sectional curvature

• Hénon–Heiles Model and Chaos
  • Conventional method
  • Evidence of chaos in a geometrical aspect

• Geometry and Chaos

• Invariants in a Generalized Model
  • Killing vector field
  • Another integrable case

• Topological Signature of Phase Transitions
  • Morse function and Euler index
  • Signatures of phase transition
  • Topological change in the mean-field XY model

It would not be an exaggeration to say that analytical mechanics for systems of point masses originated from the works of Leonhard Euler about 250 years ago [Euler]. At those times, study of fluid motions was also within his sight of mathematical formulation, and he derived fundamental equations of motion of an ideal fluid (i.e. the equation of motion and the continuity equation), which are used even now. It is remarkable that the equation of motion of an ideal fluid is represented in the same form as that of a point mass. A certain historical background on how it happened is being clarified by scholars of science history, e.g. [Tru54] or [Dar05]…

The following sections are included:

• Useful Concepts and Definitions from Mechanics
  • Action principle and Euler–Lagrange equation
  • Global invariance and conservation law
  • Fluid as continuous distribution of mass
  • Lagrangian of Galilean invariance

• Euler's Hydrodynamics
  • Equation of continuity
  • Euler's equation of motion
  • Thermodynamics and circulation theorem
  • Potential flows
  • Incompressible flows

• Conceptual Scenario of Gauge Principle
  • Examples of gauge invariance
  • Gauge invariance in quantum mechanics
  • Brief description of scenario of gauge principle
  • Gauge symmetries of fluid flows

• Equations of Lagrangian Description
  • Characterization of motion of an ideal fluid
  • Lagrangian
  • Euler–Lagrange equations
  • Noether's theorem
  • Conservation equations
  • Non-uniqueness of transformation
  • Trivial Lagrangians

• Equations of Eulerian Description
  • Eulerian representations
  • Lagrangians (translation symmetry)
  • Euler–Lagrange equation
  • Independent variations
  • Clebsch representation
  • Gauge invariance
  • Homentropic fluid

• Gauge Transformations
  • Local gauge transformations
  • Gauge invariance of D_t
  • Gauge transformation (a general formulation)

• Gauge Invariance and Differential Forms
  • Particle permutation
  • Gauge invariance in the a space
  • Representations by differential forms
  • Third trivial Lagrangian

• Rotation Symmetry
  • Rotational transformations
  • Gauge transformation
  • Covariant derivative and gauge fields

• Lagrangian Associated with Rotation Symmetry
  • Chern–Simons term
  • L_A in Eulerian space

• Improved Variational Formulation
  • Lagrangians and action principle
  • Homentropic fluid
  • Vorticity equation
  • Uniqueness of transformation
  • A new representation of Euler solution

• Significance of Helicity and L_A
  • Source of helicity
  • Invariance of helicity (Topological invariant)

• Maxwell-Type Equations in Fluid Mechanics
  • Background
  • Fluid Maxwell equations
  • Equation of sound wave
  • Lorentz force and fluid electric-field
  • Uniqueness

• Gauge Symmetries: Translation and Rotation
  • Translation symmetry
  • Rotation symmetry

• Supplement to §7.8: Vorticity as a Gauge Field
  • Particle acceleration
  • Vorticity is a gauge field

• Supplement to §7.9: Vorticity Equation and L_A
  • Vorticity equation
  • L_A in Eulerian space

• Summary

The following sections are included:

• Fundamental Concepts
  • Volume-preserving diffeomorphisms
  • Tangent vector field
  • Boundary conditions
  • Right-invariant metric
  • Right invariance of connection and curvature tensor

• Basic Tools
  • Commutator
  • Divergence-free connection
  • Coadjoint action ad
  • Formulas in ℝ³ space

• Geodesic Equation

• Jacobi Equation and Frozen Field

• Interpretation of Riemannian Curvature
  • Flat connection
  • Pressure gradient as an agent yielding curvature
  • Instability in Lagrangian sense
  • Time evolution of Jacobi field
  • Stretching of line elements

• Flows on a Cubic Space (Fourier Representation)

• Lagrangian Instability of Parallel Shear Flows
  • Negative sectional curvatures
  • Stability of a plane Couette flow
  • Other parallel shear flows

• Steady Flows and Beltrami Flows
  • Steady flows
  • A Beltrami flow
  • ABC flow

• Supplements
  • Right invariance of metric in §8.1.4
  • Arnold's formula,
  • Second proof: Explicit representation of

The following sections are included:

• A Vortex Filament

• Filament Equation

• Basic Properties
  • Left-invariance and right-invariance
  • Landau–Lifshitz equation
  • Lie–Poisson bracket and Hamilton's equation
  • Metric and loop algebra

• Geometrical Formulation and Geodesic Equation

• Vortex Filaments as a Bi-Invariant System
  • Circular vortex filaments
  • General vortex filaments
  • Integral invariants

• Killing Fields on Vortex Filaments
  • A rectilinear vortex
  • A circular vortex
  • A helical vortex

• Sectional Curvature and Geodesic Stability
  • Killing fields
  • General tangent field X

• Central Extension of the Algebra of Filament Motion

It is well known that some soliton equations admit certain geometric interpretation. An oldest example is the sine–Gordon equation on a pseudospherical surface in ℝ³ [Eis47]. The Gauss and Mainardi–Codazzi equations (§2.4) of the differential geometry of surfaces in ℝ³ yield the sine–Gordon equation when the Gaussian curvature is constant with a negative value. On the basis of the surface geometry, the Bäcklund transformation can be explained as a transformation from one surface to another in ℝ³. Namely, the relation between the new and old surfaces is nothing else than the so called Bäcklund transformation. Both were already known before the development of modern soliton theory. This will be described in Chapter 10…

This Chapter 10, §11.3.4 and §11.5 are regarded as some applications of the formulation of Chapter 2 for surfaces in ℝ³.

The following sections are included:

• Basic Ideas

• Pseudospherical Surfaces: SG, KdV, mKdV, ShG

• Spherical Surfaces: NLS, SG, NSM
  • Nonlinear Schrödinger equation
  • Sine–Gordon equation revisited
  • Nonlinear sigma model and SG equation
  • Spherical and pseudospherical surfaces

• Bäcklund Transformations Revisited
  • A Bäcklund transformation
  • Self-Bäcklund transformation

• Immersion of Integrable Surfaces on Lie Groups
  • A surface ∑² in ℝ³
  • Surfaces on Lie groups and Lie algebras
  • Nonlinear Schrödinger surfaces

• Mapping of Integrable Systems to Spherical Surfaces

The following sections are included:

• Appendix A Topological Space and Mappings

• Appendix B Exterior Forms, Products and Differentials

• Appendix C Lie Groups and Rotation Groups

• Appendix D A Curve and a Surface in ℝ³

• Appendix E Curvature Transformation

• Appendix F Function Spaces L_p, H^s and Orthogonal Decomposition

• Appendix G Derivation of KdV Equation for a Shallow Water Wave

• Appendix H Two-Cocycle, Central Extension and Bott Cocycle

• Appendix I Galilean-Invariant Lagrangian

• Appendix J Principle of Gauge Invariance

• Appendix K Pfaffian System, and Frobenius Integration Theorem

• Appendix L Orthogonal Coordinate Net, and Lines of Curvature

• References

• Index