Appendix A THE CALCULUS OF VARIATIONS Introduction The calculus of variations provides the mathematical foundation for the study of analytical mechanics as well as a number of other branches of physical sciences. In this appendix we study the essentials of the calculus of variations for applications to problems of analytical dynamics. We begin our study by introducing the notion of a functional and by way of motivation pose a number of classical problems concemed with the extremum of func­ tionals. We then show that the extremum conditions of a functional appear as solu­ tions of a differential equation - the so-called Euler-Lagrange equation. Next we con­ sider the extremization of functionals subject to different types of constraints and show that the Lagrange multiplier method provides a versatile technique for taking account of such constraints. Finally we consider functionals with variable end points and derive associated extremum conditions. A.l Functions and Functionals A function, for example u = u (x ,y), establishes a relationship between two or more variables e.g. u, and x and y . Generally the dependent variable, u, is expres sed in terms of one or more independent variables, in this case x and y, and when the numerical values of the independent variables are specified, that of the dependent vari­ able can be calculated. A functional establishes a relationship between one variable and several functions. Consider, for instance, s = fX 2 F (x, u(x), u' (x), v(x), v'(x» dx (A.l.1) XI Here in order to calculate a numerical value for s one must specify the functions u (x), and v(x). 310 For only one function, we may write X 2 s = J F (x, y (x ), y' (x )) dx (A. 1.2) XI y / Y I 2 B Yl ........ / A x O a b Figure A.I. Distance between two points in a plane Example A.l.l What is the shortest distance between two points in a plane? The distance between points A and B may be expressed as (see Figure A.I) s = J ds (a) But (b) Thus (c) Clearly when y, as a function of x, takes different forms, the functional s will take different values. Hence in equation (c) we have the integrand F = ~, and we ask which particular form of y , as a function of x , minimizes s . The study of finding the minima andlor maxima of functionals is referred to as the cal­ culus of variations. We shall develop procedures for finding such extremum condi­ tions of functionals in due course. For example A.1.1 at hand it is worth noting that: 1) Different forms of y which compete to minimize s in (c) are aH functions of x, i.e. the independent variable is not altered. 311 2) AU forms of y which are allowed to compete for minimization of s should pass through points A and B, where x = a and x = b respectively. 3) The forms of y considered must be continuous in the interval AB so that y in the integrand should remain finite. Thus there are some restrictions to be imposed on competing functions. In other prob­ lems the restrictions may take different forms. Such restrictions on the competing functions are referred to as the admissibility requirements. Problem A.I.I The brachystochrone problem: A particle m subject to gravitation g on a fric­ tionless path s of arbitrary shape z = z (x), is to move between points (0,0) and (x 2, z 2) in the shortest time possible. Determine the integrand F in X2 t=fFdx o x O x z !g Z2 .......... ........ Z A.2 Review of Extremum Values of Functions The maximum (minimum) points of functions are distinguished by the property that slight changes of the independent variables will result in a decrease (increase) in the value of the dependent variable whether the changes in the independent variable are positive or negative. For instance for a function such as y = y (x), if there exists a minimum point at x = o, we have that y (o + E) - Y (a) > O (A.2.1) Where E is a small change in x and it may be positive or negative. 312 Let us expand y (a + E) by TayIor series near the point x = a : y(a + E) = y(a) + y , Ia E + TI1" y Ia E2 + 3!1 Y '" I a E3 + (A.2.2) The derivatives y', y" etc. are alI evaluated at x = a and hence the function y = y (x) and its derivatives must be continuous at x = a. From equation (A.2.2) it can be seen that if the sign of [ y (a + E) - Y (a) ] is to be independent of the sign of E, as E approaches zero, the coefficient of y' must vanish since the sign of this term does depend upon the sign of e. Further, it is apparent from equation (A.2.2) that at a maximum point the coefficient of e2, nameIy ;! y", must be non-positive while at a minimum point it must be non-negative. Thus summarizing y' = O and y" < O maximum point (A. 2.3) y = O and y" > O minimum point It is of course possible for y' to vanish at a point where y" = O. In that case condi­ tions in relations (A.2.3) are not sufficient to determine the nature of the point and one must examine the coefficients of higher order terms. Thus at x = O of the function y = x 3 the coefficient of e3 does not vanish and since the value of this term will depend upon the sign of E one must conclude that x = O is neither a maximum nor a minimum point of the function. On the other hand x = O for the function y = x 4 can be readiIy seen to be a minimum point. It is to be noted that our definition of a maximum or minimum point, in effect describes a "tuming point" of the function. Now a given function may possess several tuming points in its range and while the conditions stated in relations (A.2.3) bold IocalIy at every tuming point, they do not yield information as to whether a given point is the global maximum or the global minimum point of the function. Thus for determining global extremum points a systematic search must be carried out amongst alI the tuming points of the function in the domain of the independent variabIe. It is also possible for a function to attain its extremum value at the boundaries of its range. In that case the extremum point need not be a tuming point and such conditions in relations (A.2.3) may not hold. In the case of a function of two independent variables such as z = z(x,y) (A.2.4) we have a surface which can be pictured as a terrain. We are now interested in finding the peaks of the hills and the Iowest points of the valIeys in this terrain. At a point x = a, y = b the following possibilities may arise i) Maximum Then z (a + E, b + a) - z (a, b) < O 313 independent of the sign of small increments e and a and as e ~ O and a ~ O. ii) Minimum Then z (a + e, b + a) - z (a , b) > o independent of the sign of e and a and as e ~ O and a ~ O. iii) A saddle point (see Figure A.2) z y Figure A.2. A saddle point Then for some values of e and a, again independent of their signs, we may have z (a + e, b + a) - z (a, b) > O z (a + e, b + a) - z (a , b) = O z (a + e, b + a) - z (a , b) < O as e ~ O and a ~ O. Again let us expand z (a + e, b + a) br. a Taylor serirS about x = a, y = b. , z (a + e, b + a) = z (a , b) + z 'x e + z 'y a + a,b a,b 'xx 2 'yy 2 'xy 'xxx 3 2\. z Ia,b e + 2\. z Ia,b a + z leaa,b + 3\. z Ia,b e + ... (A.2.5) where, ( )'x implies partial differentiation with respect to x etc., and once again alI the partial derivatives are evaluated at x = a, y = b. Hence it is assumed that z (x , y) is continuous and possesses continuous derivatives at (a, b). 314 By a similar reasoning for the independence of the sign of [z (a + e, b + a) - z (a , b)] from the signs of e and a we conclude that for the cases i, ii, and iii, cited above, we must have z'x = O Z'y = O at (a, b) (A.2.6) To find out whether the point (a, b) is a maximum or minimum point we examine the second order terms. These form a quadratic function Q (e, a) which we write as 1 [ z 'Xl: z 'xy 1[ e 1 Q (e, a) = 2![e a] z 'xy z 'yy a (A.2.7) For a maximum or minimum point the sign of Q must be independent of the signs of e and a. Thus for a minimum point, Q should remain non-negative for alI real values and either signs of e and a. Likewise, for a maximum point, Q should remain non­ positive for alI real values and signs of e and a. This property of Q, namely possessing only one sign, independent of signs of e andlor a, depends upon the form of the symmetric coefficient matrix in equation (A.2.7).