Discrete Hamiltonian Systems

1. Front Matter

   Pages i-xiv

   PDF

2. Second Order Scalar Difference Equations

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 1-44

3. Continued Fractions

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 45-69

4. Symplectic Systems

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 71-151

5. Discrete Variational Theory

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 153-197

6. Symmetric Three Term Recurrence Relations

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 199-262

7. Discrete Riccati Equations for Three Term Recurrences

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 263-294

8. Green’s Functions for Nonhomogeneous Second Order Difference Equations

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 295-317

9. Disconjugacy Criteria

   • Calvin D. Ahlbrandt, Allan C. Peterson

   Pages 319-330

10. Discrete Linear Hamiltonian Systems

    • Calvin D. Ahlbrandt, Allan C. Peterson

    Pages 331-356

11. Back Matter

    Pages 357-375

    PDF

This book should be accessible to students who have had a first course in matrix theory. The existence and uniqueness theorem of Chapter 4 requires the implicit function theorem, but we give a self-contained constructive proof ofthat theorem. The reader willing to accept the implicit function theorem can read the book without an advanced calculus background. Chapter 8 uses the Moore-Penrose pseudo-inverse, but is accessible to students who have facility with matrices. Exercises are placed at those points in the text where they are relevant. For U. S. universities, we intend for the book to be used at the senior undergraduate level or beginning graduate level. Chapter 2, which is on continued fractions, is not essential to the material of the remaining chapters, but is intimately related to the remaining material. Continued fractions provide closed form representations of the extreme solutions of some discrete matrix Riccati equations. Continued fractions solution methods for Riccati difference equations provide an approach analogous to series solution methods for linear differential equations. The book develops several topics which have not been available at this level. In particular, the material of the chapters on continued fractions (Chapter 2), symplectic systems (Chapter 3), and discrete variational theory (Chapter 4) summarize recent literature. Similarly, the material on transforming Riccati equations presented in Chapter 3 gives a self-contained unification of various forms of Riccati equations. Motivation for our approach to difference equations came from the work of Harris, Vaughan, Hartman, Reid, Patula, Hooker, Erbe & Van, and Bohner.

• Matrix
• Matrix Theory
• Microsoft Access
• Optimal control
• boundary element method
• chemistry
• continued fraction
• difference equation
• form
• functions
• hamiltonian system
• presentation
• theorem
• variable
• variational problem

`In short, the book is well written and includes all the recent results in this are. It will be very useful to undergraduate and postgraduate students in mathematics as well as to researchers in discrete integrable systems.'
Mathematical Reviews, 98m

• University of Missouri, USA

  Calvin D. Ahlbrandt

• University of Nebraska, USA

  Allan C. Peterson

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