Differential Calculus Essay

EGIM05 Nonlinear Continuum Mechanics

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DAVE N. (918636)

1. Variational Calculus

1.1 Introduction

The desire for optimality perfection is inherent in humans. The search for extremes inspires mountaineers, scientists, mathematicians and the rest of the human race. The development of Calculus of Variation was driven by this noble desire. Variational Calculus is the branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible. That integral is technically known as a functional. Many problems of this kind are easy to state, but their solutions often entail difficult procedures of differential calculus. In turn these typically involve ordinary differential equations (ODE) as well as partial differential equations (PDE).

The isoperimetric problem —that of finding, among all plane figures of a given perimeter, the one enclosing the greatest area —was known to Greek mathematicians of the 2nd century BC. The term has been extended in the modern era to mean any problem in variational calculus in which a function is to be made a maximum or a minimum, subject to an auxiliary condition called the isoperimetric constraint, although it may have nothing to do with perimeters. For example, the problem of finding a solid of

These are called variational principles and are usually expressed by stating that some integral is a maximum or a minimum. Minimization problems that can be analysed by the calculus of variations serve to characterize the equilibrium configurations of almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics, string theory, and many others. Many geometrical configurations, such as minimal surfaces, can be conveniently formulated as optimization