Calculus of Variation
Chap-04
B.V.Ramana
August 30, 2006
10:13
Chapter
4
Calculus of Variations function of the independent variable y(x), which is a function. Thus
L{y(x)} =
Y cn c2 A O c1 X B x2 x1
4.1 INTRODUCTION Calculus of variations deals with certain kinds of “external problems” in which expressions involving integrals are optimized (maximized or minimized). Euler and Lagrange in the 18th century laid the foundations, with the classical problems of determining a closed curve in the plane enclosing maximum area subject to fixed length and the brachistochrone problem of determining the path between two points in minimum time. The present day problems include the maximization of the entropy integral in third law of thermodynamics, minimization of potential and kinetic energies integral in Hamilton’s principle in mechanics, the minimization of energy integral in the problems in elastic behaviour of beams, plates and shells. Thus calculus of variations deals with the study of extrema of “functionals”. Functional: A real valued function f whose domain is the set of real functions {y(x)} is known as a functional (or functional of a single independent variable). Thus the domain of definition of a functional is a set of admissible functions. In ordinary functions the values of the independent variables are numbers. Whereas with functionals, the values of the independent variables are functions. Example: The length L of a curve, c whose equation is y = f (x), passing through two given points A(x1 , y1 ) and B(x2 , y2 ) is given by
L=
x2 x1
1 + y 2 dx
Fig. 4.1
defines a functional which associates a real number L uniquely to each y(x) (the independent variable). Further suppose we wish to determine the curve having shortest (least) distance between the two given points A and B, i.e., curve with minimum length L. This is a classical example of a variational problem in which we wish to determine, the particular curve y = y(x) which minimizes the functional
B.V.Ramana
August 30, 2006
10:13
Chapter
4
Calculus of Variations function of the independent variable y(x), which is a function. Thus
L{y(x)} =
Y cn c2 A O c1 X B x2 x1
4.1 INTRODUCTION Calculus of variations deals with certain kinds of “external problems” in which expressions involving integrals are optimized (maximized or minimized). Euler and Lagrange in the 18th century laid the foundations, with the classical problems of determining a closed curve in the plane enclosing maximum area subject to fixed length and the brachistochrone problem of determining the path between two points in minimum time. The present day problems include the maximization of the entropy integral in third law of thermodynamics, minimization of potential and kinetic energies integral in Hamilton’s principle in mechanics, the minimization of energy integral in the problems in elastic behaviour of beams, plates and shells. Thus calculus of variations deals with the study of extrema of “functionals”. Functional: A real valued function f whose domain is the set of real functions {y(x)} is known as a functional (or functional of a single independent variable). Thus the domain of definition of a functional is a set of admissible functions. In ordinary functions the values of the independent variables are numbers. Whereas with functionals, the values of the independent variables are functions. Example: The length L of a curve, c whose equation is y = f (x), passing through two given points A(x1 , y1 ) and B(x2 , y2 ) is given by
L=
x2 x1
1 + y 2 dx
Fig. 4.1
defines a functional which associates a real number L uniquely to each y(x) (the independent variable). Further suppose we wish to determine the curve having shortest (least) distance between the two given points A and B, i.e., curve with minimum length L. This is a classical example of a variational problem in which we wish to determine, the particular curve y = y(x) which minimizes the functional