INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES C 0 U R S E S AND LECTURES - No. 65 1 9. Juni 1975 DIETER BESDO TECHNICAL UNIVERSITY OF BRUNSWICK EXAMPLES TO EXTREMUM AND VARIATIONAL PRINCIPLES IN MECHANICS SEMINAR NOTES ACCOMPANING THE VOLUME No. 54 BY H. LIPPMANN COURSE HELD AT THE DEPARTMENT OF GENERAL MECHANICS OCTOBER 1970 UDINE 1973 SPRINGER-VERLAG WIEN GMBH This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Copyright 1972 by Springer-Verlag Wien Originally published by Springer-Verlag Wien New York in 1972 ISBN 978-3-211-81230-3 ISBN 978-3-7091-2726-1 (eBook) DOI 10.1007/978-3-7091-2726-1 P R E F A C E The following examples to extremum and var iational principles in mechanics were delivered in a seminar which accompanied a lecture course of Profe~ sor Horst LIPPMANN~ Brunswick. Therefore~ the exam ples cannot stand for themselves~ their main function was to illustrate the results of the lecture course and to demonstrate several interesting peculiarities of the single solution methods. The problems are normally chosen to be quite simple so that numerical computations are not necessary. Nevertheless~ sometimes~ the calculations will only be mentioned and not worked out here. The sections of the seminar-course are not identical with those of the lecture course. Especial ly~ there are no examples to more or less theoretical sections of the lectures. Because of the close aonnea tion to the lectures~ no separate list of references is given. Also the denotation is mostly the same as in the lecture-notes. I say many thanks to Professor Horst LIPP MANN for his help during the preparation-time and to the International Centre for Mechanical Sciences for the invitation to deliver this seminar. Brunswick~ October 3I~ I9?0 Dieter Besdo 1. EXTREMA AND STATIONARITIES OF FUNCTIONS 1.1. Simple problems (cf. sect. l. 2 of the lecture -notes) In this sub-section, several simple problems have to demonstrate definite peculiarities which may occur if we want to calculate extrema of functions. Problem I. I. -I : Given a function f in an unlimited region f = 10 X + 12 X 2 + 12 i:! 2 - 3 X 3 - 0u :X: 2 ~- 9 X~ 2-:."'.~3 . Find out the extrema. This problem has to illustrate the application of the necessary and the sufficient conditions for extrema of func- tions. At first, we see that f is not bounded : If ~ = 0 and :x; tends to infinity we see X- +OO f-- 00 :X:- -oo f- +OO Thus, there is no absolute extremum. To find out relative extrema, we have to use the de- rivatives 6 l.Extrema and Stationarities of FUnctions ------------------· --------- of of C)?.f f,x!!!!! ox f,~ = f,X\! :: d~;i UXO'd. C)2.f a~f f : - f,x.x a () X 2 >\j.\1. ()~2 f,x\1. = - 18 (X + ';l ) , f,~';l = 2.4 - 18 (X + 'J) . Necessary condition for an extremum of a continually differentiable function is stationarity : f,JC = 0 ' f = 0. •I& This yields the two points : s ~1 = 15/24 f' = 725 In, 1 8 "' 3 1 f X.z = --8 ~ 2 = 1 z4 2. = -139 /72 . Character of Stationarities 7 ----------------------------------- But we do not know whether these points represent rel ative extrema. We examine the matrix taken at If it is positive or negative definite we have a minimum or a maximum resp., if it is positive or negative semidefinite we possibly rmy have a minimum or maximum resp., but then 1 we cannot be sure. If t>2f ax~ is not semidefinite we have no extremum but a saddle -point. Applying this we see : point 1 We check the definiteness by a direct method. We in- traduce the vector "11 = (o: f.J), then <a = i1 A < 11 > is ex- amin~d: We see 9-1 = - 72 g.1 = + 48 ex = -fi -1. A 1 is not definite or semidefinite: point I is a saddle- point. 8 l.Extrema and Stationarities of FUnctions point 2 This yields <J 2 = 11 A 2. < 71 > = 2 4 ( C( 2 + f.> 2) + 6 ( C( +f.>) 2 > 0 ~ f ex :;. 0 or .f.> :1= 0 . Hence, A 2 is positive definite, point 2 represents a :c·elative minimum. The function f has only one minimum and no maximun~. This is possible if it has the form which is sketched in Fig. 1.1.-1. f Fig. I . 1_:1_ Boundary-Extrema 9 Problem l.l-2: Given the function f = 3 x.'+ 4 ~2. + 2:2 declaired in the region G where ~ a X.2+ \1.2 + 7. 2- 1 ~ 0 (unit sphere), calculate the extrema, also boundary-extrema. This problem has to show the curious effect that ex- trema can be lost if we are not careful enough when calculat- ing extrema on boundaries. First we try to find out extrema in the interior of the region G : f = 8 u f,:z;=ZZ. ·~ a Hence, f,x. = 0 leads to • Because of f :!!: 0 "!:his must be a minimum. There is no second extremum in the interior. Inside of G, f is bounded. So there must be a maxi mum on the boundary. The boundary is described by 9. = 0. Therefore, 2: 2 = 1 - X 2 - ~ 2 can be put into f instead of Z 2• So we get :& B the function f as a function f = f (x.14) which is valid on the boundary: B B a f = f = 0 yields ~ = \i "" 0 , f = f = 1 which is a mini >X I~ ll mum of f and, therefore, cannot be a maximum 10 !.Extrema and Stationarities of FUnctions of f . Two questions arise now 1. Is x, = lJ. = 1 , f = 1 z = :t 1 a minimum of the function f in G ? The allowed region is given by ~ ~ 0 . Thus, the gradient 9rad ~ represents a vector directed towards the out side of G . Then (grad f) · ( <ilrad ~) ~ 0 is a necessary condition for a minimum or a maximum on the boundary. n . ( ( ~rad srad. ~) ~ 0 at a special point X· t is sufficient, if there is a minimum or maximum, resp., on the boundary. (~rad <&) · (~rad. f) = 12 :x.a + 16 ~2 + 4z2 > 0 shows that we can find out only maxima on the boundary. 2. We have found out only two stationary points on the boundary. They did not represent maxima. But there must be a maxi- mum. It seems to be lost. What is the reason? We have lost the maximum because of the following B mistake which we made : When establishing the function f , we did not notice that Z 2. must be positive. This would have