This course introduces calculus of variations for a comprehensive understanding of the subject and enables the student understand mechanics from this viewpoint. It also provides basic understanding of functional analysis for rigorous appreciation of engineering optimization. After taking this course, the student will be able to formulate many problems in mechanics using energy methods. The course also reinforces the understanding of mechanics and gives hands-on experience for using variational methods. Matlab programs are part of the course.
Week
Topics
1.
Classification of optimization problems and the place of Calculus of Variations in it.
Genesis of Calculus of Variations
Formulation of Calculus of Variations problems in geometry and mechanics
2.
Unconstrained minimization in n variables
Constrained minimization: KKT conditions
Sufficient conditions for constrained minimization
3.
Mathematical preliminaries: function, functional, metrics and metric space, norm and vector spaces
Banach space, Cauchy sequence; function spaces, Inner product spaces; inner product; Hilbert space; Sobolev and Lebesgue norms; continuous and linear functionals.
First variation of a functional; Gâteaux variation; Frechét differential; and variational derivative
4.
Fundamental lemma of calculus of variations and Euler-Lagrange equations
5.
Extension of Euler-Lagrange equations to multiple derivatives and multiple functions in a functional.
Calculus of variations in functionals involving two and three independent variables.
6.
Variational (energy) methods in statics; principles of minimum potential energy and virtual work
Global constraints in calculus of variations.
Local (finite subsidiary) constrains in calculus of variations.
7.
General variation of a functional; transversality conditions. Broken extremals; Wierstrass-Erdmann corner conditions
Variational methods in dynamics: Hamilton’s principle; D’Lambert principle
Invariants of Euler-Lagrange equations and canonical forms; Noether’s theorem
Minimum characterization of Sturm-Liouville problems; Rayleigh quotient for natural frequencies and mode shapes of elastic systems
8.
Stability analysis and buckling using calculus of variations
Formulating the extremization problem starting from the differential equation; self-adjointness of the differential operator; and methods to deal with conservative and dissipative system
Multi-variable calculus and familiarity with Matlab
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