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: TThF 3:00-4:00pm FB370

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 Books

  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.

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

Useful links

Wiki, Menger, Mathworld, Derivative, Semi-coninuity


Course Summary

This is second consecutive course related to the subject of the calculus of variations (the first dealt with first variation) in this year-long series of two courses. However, both courses are independent of each other. In this course, second variation will be studied mainly and a framework for direct methods in the calculus of variations will be developed as well as some crucial modern advances related to hyperelasticity will be discussed. We will, exclusively, study the questions related to local and global minimizers (not just an extremal). We will elaborate upon this subject, mainly, in the context of three-dimensional hyperelasticity (theory of elasticity which involves a strain energy density function) and specific examples will be also taken from hyperelasticity in one- and two-dimensions.

  1. (i)Definitions: first variation, Euler equation, second variation, strong/weak minimizer.

  2. (ii)Necessary condition for a minimum, Legendre condition, Jacobi condition, conjugate points.

  3. (iii)Sufficient condition for a minimum, Weierstrass E-function. Field of extremals.

  4. (iv)Definitions: convexity, quasiconvexity, strong ellipticity, weak derivative, minimizing sequences, lower semi-continuity, relevant notions specific to three dimensional theory of hyperelasticity. Interpretation of convexity and strong ellipticity in the context of hyperelasticity. Singular minimizers, Lavrentiev phenomenon and their relation to some problems in mechanics.

  5. (v)Conclusion of the course with a discussion of some open problems.

Course flyerME624_2008_files/ME624.pdfshapeimage_8_link_0

Minimizers in Mechanics and Elasticity

Some information on Lectures

  1. (I)Lecture 0: (31 Dec) Quiz for certain preliminaries

  2. (II)Lecture 1: (3 Jan) Overview of Course

  3. (III)Lecture 1-8 (8): (3, 4, 8, 10, 11, 15, 17, 18 Jan) Local Minimum of a Function defined on a Euclidean space, Local Minimum of a Functional defined on a Banach space, First Variation and Second Variation, Discussion on Necessary Condition and Sufficient Condition for a Local Minimizer in One Dimensional Variational Problem and Higher Dimensional Variational Problem, C1 Extremal and C2 Extremal, Euler equation, C0 local minimizer and C1 local minimizer

  4. (IV)Lecture 9-18 (10): (22, 24, 25, 29 Jan, 5, 7, 8, 12, 14, 19 Feb) 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

  5. (V)Lecture 19-34 (15): (20, 20, 21, 22, 26, 28, 29 Feb, 4, 6, 7, 13, 25, 27, 28 Mar, 1 Apr) 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)

  6. (VI)Lecture 35-43 (8): (3, 4, 8, 10, 11, 15, 17, 18 Apr) Definitions: Convexity, QuasiConvexity, Rank-one Convexity, 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. Discussion on the Direct Methods in the Calculus of Variations and their influence on some developments in the Theory of Elasticity. Singular minimizers, Lavrentiev phenomenon and their relation to some problems in mechanics.