Unit 1

Calculus of Variations: Maxima and minima – The simplest case – Illustrative examples – Natural boundary conditions and transition conditions – Concept of functional with simple example – Variation of a functional (only necessary conditions) – Simple variational problem – Euler equation – The more general case of variational problems – Constraints and Lagrange multipliers – Variable end points.

Unit 2

Sturm-Liouville problems – Hamilton’s principle – Lagrange’s equations – Generalized dynamical entities – Constraints in dynamical systems – Applications in dynamics of particles, vibrating string, vibrating membranes, theory of elasticity – The variational problem of a vibrating elastic plate – Direct methods in calculus of variations – The Rayleigh-Ritz and finite difference methods. (Book-1)

Unit 3

1. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications, 2000
2. Advanced Engineering Mathematics, E Kreyszig, John Wiley and Sons, Ninth Edition, 2012

• A. S. Gupta, calculus of Variations with Applications, Prentice Hall of India, 1997
• M. K. Venkatraman, Advanced Engineering Mathematics, 2010.
• M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerical methods for Scientific and Engineering Computation, New Age International Publishers, Fifth ed., 2007.
• Steven Chapra and Raymond Canale, Numerical Methods for Engineers, McGraw Hill, 2007.