ME 624




Course details

Course No: ME 624

Instructor: Basant Lal Sharma

Department: Mechanical Engineering

Units: (L-T-P-D-U) 3-0-0-0-4

Schedule: LEC: MWF 9:00-10:00TB210

Prerequisite: Basic courses in Analysis (Sequences and Limits) and Calculus (Differentiation and Integration)

Note: The course is offered and structured as an elective for postgraduate students and advanced undergraduate students. Anyone else, who is interested, may take the permission of instructor.


Email: bls at

Phone: 6173

Main Book

  1. 1.Gelfand, I. M.; Fomin, S. V., 1963. Calculus of variations. Prentice-Hall, (Textbook).

  2. 2.Giaquinta, M., and Hildebrandt, 2004. S. Calculus of variations I, Springer-Verlag.

  3. 3.Dacorogna, B., 1989. Direct methods in the calculus of variations. Springer.

  4. 4.Braides, A. 2002. Gamma convergence for beginners. Oxford university press.

Useful links

Wiki, Menger, Mathworld, Derivative


Course Summary

In this course, the framework of calculus of variations will be introduced and developed after an introduction to elementary functional analysis and the concept of so called functional. First the necessary condition for a extremum (via so called first variation) will be dicussed. The Euler equation will be derived for some simple problems in mechanics and physics and solutions of such problems will be studied in a few cases. Then we will study the questions related to local and global minimizers (and not just an extremal, via the so called second variation) and direct methods in the calculus of variations, respectively. Lastly some crucial modern advances in the calculus of variations will be discussed. The contents in more detail are as follows:

  1. History of calculus of variations: Discussion of certain classical and modern problems in mechanics that led the emergence and development of calculus of variations. Contributions of Bernoulli(s), Euler, Lagrange, Jacobi, Weierstrass, Hamilton, Legendre and some others will be discussed briefly.

  2. Definition of a function, function space (with examples of function spaces that are important in engineering and sciences) and so called ‘functional’.

  3. Definition of the first variation of a functional and a variational derivative. Necessary condition for an extremum. Euler equation. Discussion on Null Lagrangian and natural boundary condition. Constraints and Lagrange multipliers (with examples from classical mechanics).

  4. Noether theorem and conservation laws, Weierstrass-Erdmann condition and more general jump conditions (with examples from continuum mechanics of bodies with defects).

  5. Definition of the second variation and the strong/weak minimizer. Necessary condition for a minimum, Legendre condition, Jacobi condition, conjugate points. Sufficient condition for a minimum, Weierstrass E-function. Field of extremals.

  6. Definitions of convexity, quasiconvexity, strong ellipticity, weak derivative, minimizing sequences, lower semi-continuity. Some relevant notions specific to three dimensional theory of hyperelasticity, interpretation of convexity and strong ellipticity in the context of hyperelasticity.

  7. Singular minimizers, Lavrentiev phenomenon, Gamma convergence and their relation to some problems in mechanics.

  8. Conclusion of the course with a discussion of some open problems.

Course FlyerME624_2012_files/ME624.pdfshapeimage_7_link_0

Some information on Lectures

  1. (I)Lecture 1 (1) : Overview of Course, History of calculus of variations.

  2. (II)Lecture 2-4 (3): Function spaces and Introduction to Functional Analysis: Sets, Mappings and some Function Spaces with useful Structure, Differentiation and Integration, Functional Derivative.

  3. (III)Lecture 5 (1): Extremum of a Function defined on a Euclidean space, Local Minimum of a Function defined on a Euclidean space, Extremum of a Functional defined on a Banach space, Local Minimum of a Functional defined on a Banach space, First Variation and Second Variation.

  4. (IV)Lecture 6-11 (6): Weak Extremal, Euler equation, Natural Boundary conditions , Null Lagrangian, Characterization of Null Lagrangians, Variational Problems of Higher Order.

  5. (V)Lecture 12-14 (3): Constraints and Lagrange multipliers, Isoperimetric constraints, Holonomic constraints, Non-Holonomic constraints, Transversality conditions.

  6. (V)Lecture 15-20 (6): Noether theorem and Conservation laws, Application of Noether Theorem in One Dimension: Particle Mechanics, Application of Noether Theorem in One+One Dimension: Elastodynamics of a Hyperelastic Bar, Application of Noether Theorem in One+Three Dimensions: Elastodynamics of a Hyperelastic body, Lipschitz Extremal and Weierstrass-Erdmann Jump conditions.

  7. (VI)Lecture 21-23 (3): Discussion on Necessary Condition and Sufficient Condition for a Local Minimizer in One Dimensional Variational Problem and Higher Dimensional Variational Problem. C0 local minimizer and C1 local minimizer.

  8. (VI)Lecture 24-27 (4): Necessary Condition and Sufficient Condition for C1 local minimizer: Legendre condition in One Dimensional case, Jacobi’s Conjugacy condition in One Dimensional case, Legendre-Hadamard condition in Higher Dimensional case, Strong ellipticity and Strongly elliptic operator, Jacobi Necessary condition and Jacobi Sufficient condition based on eigenvalue of Jacobi operator.

  9. (VII)Lecture 28-32 (5): Necessary Condition and Sufficient Condition for C0 local minimizer: Weierstrass Necessary Condition in Higher Dimensional Case, Field of Curves, Field of Extremals, Mayer field, (Aside on Exterior Forms, Exterior Product, Manifolds, Differential Forms, Exterior Derivative and Pullbacks) , Weierstrass Sufficient Condition in One Dimensional Case, *Weierstrass Sufficient condition in Higher Dimensional case (*if time permits).

  10. (VIII)Lecture 33-40 (8): Definitions: Convexity, QuasiConvexity, Rank-one Convexity, Weak Derivative, Minimizing sequences, Lower Semi-Continuity. Discussion of some results involving Convexity and Strong Ellipticity in the context of Hyperelasticity. Discussion on the Direct Methods in the Calculus of Variations and their influence on some recent developments such as Gamma convergence in the Calculus of Variations. Singular minimizers, Lavrentiev phenomenon, Gamma convergence and their relation to some problems in mechanics.

Calculus of Variations