MATH 648 – Calculus Of Variations (Spring 2022)

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Text: The Calculus of Variations —by Bruce van Brunt;  Publisher: Springer  Universitext Series; Publication year 2006;  ISBN 978-0387402475;

Supplementary texts:  

For Physics, Chemistry and Engineering majors: 

  1. Variational Principles of Continuum Mechanics I (Fundamentals) -by Victor Berdichevsky, (2009), Springer;
  2. The Variational Principles of Mechanics, (4th Edition), by Cornelius Lanczos, Dover Publications; 
  3. Classical Mechanics (3rd Edition) by Goldstein, Poole and Safko, Addison Wesley Publishing Company;
  4. The Feynman’s lectures on physics  (Volume 1, the 6th printing, 1977), by Feynman, Leighton and Sands,  Addison Wesley Publishing Company.  
  5. Mechanics (3rd Edition), by Landau and Lifshitz, Published by Butterworth/Heinemann.

For Math majors:

  1. Mathematical Methods of Classical Mechanics, by V.I. Arnold, (1978), Springer;
  2. Calculus of Variations I and II, by Giaquinta and Hilderbrandt, (1996), Springer; 
  3. Calculus of Variations, by Gelfand and Fomin, Dover Publications;
  4. Methods of Mathematical Physics (Volume II), by Hilbert and Courant 
  5. Notes, papers and handouts that I will distribute over the lecture, particularly relating to modern probability theory. 

Jin Feng’s Office:

Room 510, Snow Hall; Phone: 785-864-3764;


Office hour: Tuesday Thursday afternoon by appointment, more details TBA.

Lecture time: Tuesday and Thursday. 11:00am-12:15 pm, Room 156, Snow Hall

Homework and exams:

Exercises will be assigned as we lecture through selected materials from the texts and reference books, lecture notes etc. A final will be given in the form of a reading and presentation project.  I will assign some papers/book chapters/advanced material for you to read. See a sample of topic below.  Each one of you need to deliver a presentation about 30 minutes to the whole class or to me during my office hour. You can also choose material reflecting your own research interests, but please get my approval first (to ensure level of difficulty of material is appropriate). In case of material of interests covers too large a topic to deliver in one presentation, you can pair with others, each presenting part of a coherent topic.  


Course information:  

  1. Hamilton’s principle and other variational principles from physical sciences; 
  2. Euler-Lagrange equations;
  3.  Legendre transformation and Hamilton’s equation; Symplectic maps, Hamilton-Jacobi ODE;
  4. Noether’s theorem and variational symmetries;
  5. Canonical transforms, Generating functions, and  first order Hamilton-Jacobi PDEs;
  6. Method of characteristics and Hamilton-Jacobi ODEs;
  7. Re-parametrization and parametric Variational Problems, and Huygen’s principle;
  8. Second variation and conjugate points;

(if time allows, I will select from the following topics) 

  1. Time-space variational problems and applications to continuum mechanics;
  2. Selected topics in thermodynamics;
  3. An introduction to averaging for Hamiltonian systems;
  4. Selected topics in equilibrium statistical mechanics,  large deviation and equivalence of ensembles using modern probability theory.

Suggested additional topics for reading-and-presentation projects: (This list may grow as the lectures go).

  1. Equations in Fields and Continuum Mechanics: Chapters 3,4 of Berdichevsky; Chapter 13 of Goldstein, Poole and Safko; Chapter XI of Lanczos. 
  2. Why the Lagrangian consistent with Newtonian mechanics should be kinetic MINUS the potential energies, a relativistic interpretation. Pages 42,43 of Berdichevsky. 
  3. Topics in thermodynamic formalism and averaging: Chapter 2 of Berdichevsky. 
  4. Hamilton-Jacobi theory through the point of view of Huygen’s principle: page 208-217 of Gelfand-Fomin, and Arnold page  248-258.  Complementary math material on indicatrix and figuratrix: Chapter 7.3.2 of Giaquinta and Hilderbrandt .
  5. Fermat’ principle and Snell law in Optics (a complete, conceptual, proof with no computation) Chapter 26-3 of Feynman. 
  6. Critical points and mountain pass lemma, Chapter 8.5 of L. C. Evans’ book on PDE, plus some lecture notes…
  7. The Maupertuis principle, Section 44 of Landau and Lifshitz’s Volume 1 in Course of Theoretical Physics, Mechanics.
  8. PDEs of wave type, characteristics, Hamilton-Jacobi and small wave expansions, V.P. Maslov,  page 139-185, Proceedings of the International Congress of Mathematicians, August 16-24, 1983, Warszawa. 
  9. Complete integrals, envelopes and Characteristic, Chapter 3.2 of L.C. Evans’ book Partial Differential Equations. Chapter I.4 and Chapter II of Hilbert and Courant.

Exercises from the text:

  1. Exercises 2.2 : 4 and 5; Exercises 2.3: 3.