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Syllabus ( MATH 590 )

   Basic information
Course title: Introduction to Hamiltonian Formulation of Differential Equations
Course code: MATH 590
Lecturer: Prof. Dr. Oğul ESEN
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: none
Professional practice: No
Purpose of the course: This is an introductory course on the theory of geometric mechanics. Its purpose is to introduce the Hamiltonian theory of differential equations both on linear spaces and manifolds. To this end, symplectic and Poisson vector spaces, and symlectic and Poisson manifolds will be introduced. On those spaces, the Hamilton’s equations will be presented.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Define symplectic and Poisson structures on linear spaces.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently

    Method of assessment

    1. Written exam
    2. Homework assignment
  2. Write Hamilton’s equations on Poisson and symplectic spaces

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently

    Method of assessment

    1. Written exam
    2. Homework assignment
  3. Determine differential equations admitting Hamiltonian formulations

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently

    Method of assessment

    1. Written exam
    2. Homework assignment
  4. Define symplectic and Poisson structures on manifolds

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently

    Method of assessment

    1. Written exam
    2. Homework assignment
   Contents Up
Week 1: Vector spaces, subspaces, linear transformations
Week 2: Dual spaces, multilinear functionals
Week 3: Symplectic vector spaces
Week 4: Symplectic transformations
Week 5: Hamilton’s equations
Week 6: Hamiltonian flows
Week 7: Poisson brackets
Week 8: KdV Eq., Wave Eq.,
Week 9: Schrödinger Eq., Sine-Gordon Eq. Midterm Exam.
Week 10: Manifolds
Week 11: Vector fields and differential forms
Week 12: Symplectic manifolds, Hamiltonian systems, Canonical maps
Week 13: Poisson brackets on symplectic manifolds
Week 14: The canonical Hamiltonian systems on cotangent bundles
Week 15*: -
Week 16*: Final Exam.
Textbooks and materials: • Marsden, J. E., & Ratiu, T. (1999), Introduction to mechanics and symmetry, Second
edition, Texts in Applied Mathematics 17, Springer-Verlag, New York.
• Libermann, P., & Marle, C. M. (2012), Symplectic geometry and analytical mechanics (Vol. 35), Springer.
• Abraham, R. & Marsden, J. E. (1978), Foundations of mechanics, Benjamin/Cummings Publishing Company, Reading, Massachusetts.
Recommended readings: • Arnold, V.I. (2013) Mathematical methods of classical mechanics (Vol. 60), Springer
• Holm, D.D. (2008) Geometric Mechanics. Part I and Part II, Imperial College Press, London,
• Holm, D.D., Schmah, T. & Stoica C. (2009) Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Oxford University Press,
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 9 30
Other in-term studies: 0
Project: 0
Homework: 3,6,12,14 30
Quiz: 0
Final exam: 16 40
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 3 14
Practice, Recitation: 0 0
Homework: 12 4
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 20 1
Mid-term: 3 1
Personal studies for final exam: 30 1
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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