Canonical Equations
In this chapter we will calculate the canonical equations which belong to a simple problem. As we will see from the result, they will not enable us to find out the equations of motion in a simpler way than, for example by using LAGRANGE-equations of 2nd k
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AND
LECTURES
-
No.
65
19. 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 variational 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. Especially~ 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 LIPPMANN for his help during the preparation-time and to the International Centre for Mechanical Sciences for the invitation to deliver this seminar. Brunswick~
October
Dieter Besdo
3I~
I9?0
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
-
0
2
u :X: ~-
.3 9 X~ 2-: ."'~
.
Find out the extrema.
This problem has to illustrate the application of the necessary and the sufficient conditions for extrema of functions. At first, we see that f is not bounded : If ~ = 0 and :x; tends to infinity we see X-
+OO
: X : - -oo
f--
00
f - +OO
Thus, there is no absolute extremum. To find out relative extrema, we have to use the derivatives
6
l.Extrema and Stationarities of FUnctions
------------------ยท
f,x!!!!!
of
f,~
ox f,x.x a
f,x\1. =
f,~';l
=
-
=
of
f,X\! ::
d~;i
C)2.f () X 2
---------
f
C)?.f
UXO'd.
a~f
>\j.\1.
:()~2
18 (X + ';l ) ,
2.4 - 18 (X + 'J) .
Necessary condition for an extremum of a continually differentiable function is stationarity :
f,JC
=
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