ME 698C




Course details

Course No: ME 698C

Instructor: Basant Lal Sharma

Department: Mechanical Engineering

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

Schedule: LEC: MWF 11:00-12:00pm 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 Book

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

Useful links

Wiki, Menger, Mathworld, Derivative


Course Summary

This course is the first in a sequence of two courses related to calculus of variations and attempts to develop the stage for advanced courses in this field. The contents of this first course for the duration of one semester may depend on the pace of instruction and performance of students. In this course, the framework of calculus of variations will be developed and the necessary condition for a extremum (so called first variation) will be exclusively dicussed. The Euler equation will be derived for some simple problems in mechanics and solutions of such problems will be studied in a few cases.

  1. (i)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,

  2. (ii)Weierstrass, Hamilton, Legendre and some others will be discussed.

  3. (iii)Definition of a function, function space (with examples of function spaces that are important in mechanics) and so called ‘functional’.

  4. (iv)The variation of a functional and a variational derivative. Necessary condition for an extremum. Euler equation.

  5. (v) Discussion on Null Lagrangian and natural boundary condition. Constraints and Lagrange multipliers (with examples from mechanics).

  6. (vi)Noether’s theorem and conservation laws, Weierstrass-Erdmann condition and more general jump conditions (with examples from continuum mechanics).

  7. (vii)Legendre transformation. Hamilton-Jacobi equation. Canonical formulation.

  8. (viii)Conclusion of the course with a flavour of second variation and use of direct methods in the calculus of variations.

Course flyerME698C_2007_files/ME698C.pdfshapeimage_7_link_0

Some information on Lectures

  1. (I)Lecture 0 (30 Jul): Quiz for certain preliminaries

  2. (II)Lecture 1 (1 Aug): Overview of Course

  3. (III)Lectures 2, 3 (2, 6 Aug): History of calculus of variations

  4. (IV)Lecture 4-10 (8, 10, 13, 17, 20, 21 Aug, 2 extra lectures on 21 Aug): Function spaces and Introduction to Functional Analysis: Sets and some useful Spaces, Differentiation and Integration, Functional Derivative

  5. (V)Lecture 11-20 (22, 24, 29 Aug, 3, 5, 7, 10, 12, 14, 17 Sep): Extremum of a Function defined on a Euclidean space, Extremum of a Functional defined on a Banach space, First and Second Variation, Weak Extremal, Euler equation, Weak Extremals and Extremals, Null Lagrangian, Characterization of Null Lagrangians, Natural Boundary conditions, Constraints and Lagrange multipliers, Isoperimetric constraints, Holonomic constraints, Non-Holonomic constraints, Transversality conditions

  6. (VI)Lecture 21-33 (19, 21, 24, 26, 28 Sep, 1, 3, 5, 12, 22, 24, 29, 31 Oct): Noether's 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. (VII)Lecture 34-38 (2, 5, 12, 14, 16 Nov): Legendre Transformation, Canonical Euler Equation and Canonical Transformation, Hamilton-Jacobi Equationi, Structure of the phase space.

Calculus of Variations in Mechanics