Course Content:

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; Definition of a function, function space (with examples of function spaces that are important in engineering and sciences) and so called 'functional'; 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); Noether theorem and conservation laws, Weierstrass-Erdmann condition and more general jump conditions (with examples from continuum mechanics of bodies with defects); 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; 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; Singular minimizers, Lavrentiev phenomenon, Gamma convergence and their relation to some problems in mechanics; Conclusion of the course with a discussion of some open problems

Lecturewise Breakup (based on 50min per lecture) (total lecture 40)

I. Introduction : (Lecture 1) (1 Lectures)

• Overview of Course, A brief historical survey of calculus of variations. (Handout on Brief history of the calculus of variations: two tables giving timeline of events related to classical calculus of variations.). Description of Brachistochrone vs isoperimetrical problems (more like like linear space vs nonlinear spaces (eg. manifolds), constrained problems vs unconstrained etc).

II. Introduction to Function spaces and Functional Analysis: (Lecture 2-4)(6 Lectures)

• Sets, Mappings and some Function Spaces with useful Structure, Differentiation and Integration, Functional Derivative. Mentioning Banach space and Hilbert space as well as metric spaces and, very briefly, manifolds also. Detailed definitions of C, C n , D,L p , Wk,p , etc function spaces. Some motivating examples from mechanics: elasticity, particle mechanics etc, involving such function spaces. Short review of real analysis and differentiation/integration, correspondence between variation and differential. Definition of directional derivative: Gˆateaux derivative. Highlight Gˆateaux vs Fr´echet derivative. Strong vs weak derivative.

III. (Formulation) : (Lecture 5-6) (2 Lectures)

• 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. Strong vs weak minimum.

IV. (First Variation : Euler Equation: (Lecture 7-15) (9 Lectures)

• Weak Extremal, Necessary condition for extremum: Euler equation, Null Lagrangian and its importance in mechanics (eg. elasticity), Characterization of Null Lagrangians, Some typical boundary conditions and their form in the calculus of variations: Natural Boundary conditions, Constraints and Lagrange multipliers, Isoperimetric constraints, Holonomic constraints, Non-Holonomic constraints, Transversality conditions.

V. Conservation laws and Jump conditions : (Lecture 16-21) (6 Lectures)

• Statement of Noether's theorem (without proof!). Derivation of conservation laws in mechanics, eg: Jintegral, configurational force (Eshelby), and other integrals important in elasticity. 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. Non-smooth extremals: Lipschitz Extremal and Weierstrass-Erdmann Jump conditions. Examples from solid-solid phase transition..

VI. Minimizers : (Lecture 22-24) (3 Lectures)

• Discussion on Necessary Condition and Sufficient Condition for a Local Minimizer in One Dimensional Variational Problem and Higher Dimensional Variational Problem. C 0 local minimizer and C 1 local minimizer. Necessary condition for a minimum, Legendre condition, Jacobi condition, conjugate points. Sufficient condition for a minimum, Weierstrass E-function. Field of extremals. .

VII. C¹ local minimizer : (Lecture 25-29) (5 Lectures)

• Necessary Condition and Sufficient Condition for C ¹ 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.

VII. C⁰ local minimizer : (Lecture 30-34) (5 Lectures)

• HNecessary Condition and Sufficient Condition for C ⁰local 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).

VII. Direct Methods : (Lecture 35-40) (6 Lectures)

• Definitions: Convexity, QuasiConvexity, Rank-one Convexity, Strong Ellipticity, Weak Derivative, Minimizing sequences, Lower Semi-Continuity, (if time permits: conjugate spaces and weak*-convergence). Some relevant notions specific to three dimensional theory of hyperelasticity, Discussion of some results involving Convexity and Strong Ellipticity in the context of Hyperelasticity. Singular minimizers, Lavrentiev phenomenon and their relation to some problems in mechanics and elasticity. Discussion on the Direct Methods in the Calculus of Variations and their influence on some recent developments in the field.

References:

1. Gelfand, I. M.; Fomin, S. V., 1963. Calculus of variations, Prentice-Hall, (Textbook)
2. Giaquinta, M., and Hildebrandt, 2004. S. Calculus of variations I, Springer-Verlag
3. Dacorogna, B., 1989. Direct methods in the calculus of variations, Springer.