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Math 5451:
Calculus of Variations and Tensor Calculus


(Table of Contents and Its Time Line)


PART I: CALCULUS OF VARIATIONS


Lecture 1

Chapter I: Fundamental Ideas

  1. Multivariable Calculus as a Prelude to Calculus of Variations
  2. Some Simple Problems in Calculus of Variations
  3. Methods for Solving Problems in Calculus of Variations
    1. Method of Finite Differences

      Lecture 2
    2. Method of Variations
      1. Variants and Variations
      2. Variational Derivative
      3. Euler's Differential Equation
    3. Solved Example

      Lecture 3
  4. Integration of Euler's Differential Equation

Chapter II: Generalizations

  1. Functional with Several Unknown Functions

    Lecture 4
  2. Extremum Problems with Side Conditions
    1. Isoperimetric Problems
      1. Heuristic Solution
      2. Solution via Constraint Manifold

      Lecture 5
    2. Variational Problems with Finite Constraints
  3. Variable End Point Problem
  4. Extremum Problem at a Moment of Time Symmetry

    Lecture 6

    Generic Variable Endpoint Problem

    1. General Variations in the Functional
    2. Transversality Conditions
    3. Junction Conditions

    Lecture 7
    Parametrization Invariant Problem
    1. Variational Principle for a Geodesic

      Lecture 8
    2. Equation of Geodesic Motion
      1. Parametric Invariance
      2. Parametrization in Terms of Curve Length

        Lecture 9
      3. Physical Significance of the Equation for a Geodesic
      4. Equivalence Principle and ``Gravitation''=``Geometry''

Lecture 10

Chapter III: Variational Formulation of Mechanics

  1. Hamilton's Principle

    Lecture 11
  2. Hamilton-Jacobi Theory
    1. The Dynamical Phase
    2. Momentum and the Hamiltonian
    3. The Hamilton-Jacobi Equation

      Lecture 12
    4. Hamilton-Jacobi Description of Motion: Constructive Interference

      Lecture 13
    5. Spacetime History of a Wave Packet
    6. Hamilton's Equations of Motion

      Lecture 14
    7. Phase Space of a Dynamical System
    8. Constructive Interference Implies Extremal Paths

      Lecture 15
    9. Applications
      1. Hamilton-Jacobi Equation Relative to Curvilinear Coordinates
      2. Separation of Variables
      3. Particle Motion in a Combined Dipole-Monopole Potential
      4. Motion in the Field of Two Unequal Bodies: Prolate Spheroidal Coordinates

    Lecture 16
  3. Hamilton's Principle for the Mechanics of a Continuous Medium
    1. Variational Principle
    2. Euler-Lagrange Equation

      Lecture 17
    3. Examples and Applications

Chapter IV: Direct Methods in the Calculus of Variations

  1. Minimizing Sequence
  2. Finite-Dimensional Approximation

    Lecture 18
  3. Rayleigh's Variational Principle
    1. The Rayleigh Quotient
    2. The Rayleigh-Ritz Principle
    3. Vibrations of a Circular Membrane

Lecture 19

PART II: TENSOR CALCULUS

Chapter V: Tensor Algebra

  1. Vectors vs. Covectors
    1. Construction of Linear Functionals: The Dual Basis

      Lecture 20

    2. The Differential of a Function
    3. Vector as a Derivation
    4. Dual Bases
    5. Invariance of Duality

    Lecture 21
  2. Metric on a Vector Space
    1. Metric as a Map Between Vectors and Covectors
    2. Reciprocal Vector Basis

    Lecture 22
  3. Tensors as Multilinear Maps
    1. Tensors: Their Components
    2. Tensors: Their Basis Representation
    3. Tensor Space
    4. How to Construct New Tensors
      1. Raising and Lowering Indeces
      2. Contraction of a Tensor

Lecture 23

Chapter VI: Geometrical Structures

  1. The Commutator of Two Vector Fields

    Lecture 24
  2. Parallel Transport
    1. Covariant Differential

      Lecture 25
    2. Covariant Derivative
      1. Differential of a Function
      2. Differential of a Vector
      3. Differential of a Vector Field
      4. Rules for Taking Covariant Derivatives
    3. Parallel Transport in the Euclidean Plane
    4. Covariant Derivative of a Vetor Field
    5. Covariant Derivative of a Covector Field
    6. Covariant Derivative of a Tensor Field

      Lecture 26
    7. Parallel Vector Fields
  3. Torsion
    1. Closed Parallolograms
    2. Burger's Vector

    Lecture 27
  4. Curvature
    1. Rotation of a Vector Transported Around a Closed Loop
    2. Riemann Curvature Tensor
    3. Equation of Geodesic Deviation
    4. Rotation = Relative Acceleration

    Lecture 28
  5. Metric
    1. Metric-Induced Parallel Transport
    2. Metric Compatibility via Extremal Paths

    Lecture 29
  6. Some Computational Aids
    1. Divergence of a Vector Field
    2. Gauss's Theorem




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gerlach@math.ohio-state.edu 2003-07-11