General Variation of a Functional

Transcription

1 Lecture 15 General Variation of a Functional Transversality conditions Broken etremals Corner conditions ME 256 at the Indian Institute of Science, Bengaluru G. K. Ananthasuresh Professor, Mechanical Engineering, Indian Institute of Science, Banagalore suresh@mecheng.iisc.ernet.in 1

2 Outline of the lecture Variable end conditions: motivating eamples General variation Transversality conditions Weierstrass-Erdman corner conditions What we will learn: Why we need to deal with variable end conditions in calculus of variations How to take general variation and how it affects only the boundary conditions and not the differential equation What broken etremals are How we can get the regular boundary conditions as special cases 2

3 Modified brachistochrone problem A Minimize H Now, point B can be anywhere on a given curve represented by g L B We want to find y() such that an object will reach any point on in the least time. Note that the change in the problem statement comes only in the end condition and not in the functional. 3

4 Another modification A Minimize Note again that the change in the problem statement comes only in the end conditions and not in the functional. H Now, point A can be anywhere on a given curve represented by g L B We want to find y() such that an object will reach any point on starting from any point on in the least time. 4

5 A general problem with variable end conditions What do we do when ends are not given? Recall that we had taken a variation (a perturbation) around a minimal curve y * () and equated the first-order term to zero to establish the necessary condition. Here, the perturbation should be taken for y * () and the two ends. Variable ends means that both ends can also be perturbed. That is, the domain over which we integrate is variable. In such a case, we take what is called a general variation in which ends are also perturbed. See the net slide 5

6 General non-contemporaneous variation (related to non-contemporary) Now we have perturbed not only the curve but also the ends! This type of variation is called non-contemporaneous variation. The term non-contemporaneous must be in the contet of time-related problems. We are shifting the -ais. So, y and y * are not defined on the same domain. 6

7 First-order change with general variation We got both on the same domain. So, these two terms come out separated. This is an approimation because the perturbed domains are very small. 7

8 Etensions of the domain at either end Slope at the first end Slope at the second end Differences between the original and perturbed curves at either end The domains of the original curve and the perturbed curve need to be etended as shown with blue lines by maintaining tangency to the respective curves. 8

9 The first term of the first-order term * * * * (, ) (, ) ( ) ( ) { } F y + h y + h d F y y d + F h + F h d 2 2 * * d = F( y, y ) d + Fy Fy' h d + Fy ' h d * * d F( y, y ) d Fy Fy' h d Fy' h Fy' h d ( ) ( ) ( ) = + + y 2 1 y' 2 1 A result we had derived earlier in Lecture 11; see Slides 3 and 4 in Lecture 11. 9

10 And now By substituting for this from the preceding slide 2 d J F F hd+ F h F h F + F d 2 1 Recall from slide 8: 1 ( ) ( ) ( ) ( δ ) ( δ ) y y' y' y'

11 Necessary condition and boundary conditions finally. First order is equated to zero for the necessary condition, as usual. By invoking the fundamental lemma, we get the differential equation: Boundary conditions Note that the differential equation, the Euler-Lagrange equation, did not change! Note that the boundary condition of the fied end conditions comes out neatly when the variation in the end conditions are zero. That is, when 11

12 Boundary conditions when restricted to given curves A B {( F F ) } y ' φ 1 y δ + ( ) = 0 {( F F ) } y ' φ 2 y δ + ( ) = 0 These are called transversality conditions

13 Transversality conditions Transversality has something to do with being orthogonal, i.e., perpendicular. It is indeed so for certain functionals. It means that the minimal curve is orthogonal to the boundary curve! 13

14 Transversality and brachistochrone A The optimal curve is perpendicular to the two given curves at either end. B Even though the transversality is limited only to special form of the functional, the name stuck for all types of functionals. What is in a name, anyway? 14

15 Eample: beam guided at one end 1 ( ) 2 F = EI w qw 2 Min w( ) L 1 ( ) 2 J = EI w qw d 2 0 because {( F F ) } w' φ 2 w δ 2 + ( ) = 0 But there is no F w term here. So, we need to derive the transversality condition for w term. 15

16 Transversality condition for yʹʹ term Resume from Slide 10 by including y term. From Slide 10 of this lecture From Slide 17 in Lecture 11 16

17 Etended transversality conditions By invoking the fundamental lemma, we get the differential equation: Boundary conditions Note that the differential equation, the Euler- Lagrange equation, did not change, once again! It does not in all cases when the end conditions change. Note that the boundary condition of the fied end conditions comes out neatly when the variation in the end conditions are zero. That is, when 17

18 Etended transversality conditions (contd.) A B 18

19 Back to the guided beam because 19

20 For two functions in one variable With variable end conditions Transversality conditions Differential equations do not change, as usual. 20

21 Minimal curves need not be smooth! So far, we had assumed that minimum curves are smooth, i.e., the slope of y is continuous. But what if it is not? We get a kink or a sudden bend in the curve. Such etremal curves are called broken etremals. They happen in problems where something in the integrand of the function suddenly changes. In such a case, variable conditions equations come to rescue us. 21

22 Broken etremal conditions For the two parts for one on the right side and the other on the left side. So 22

23 Weierstrass-Erdmann corner conditions So, whenever the intermediate point is variable are continuous at the intermediate corner point. 23

24 Broken (non-smooth) etremals Recall from Slide 3 of Lecture 2 This historically first calculus of variations problem has a non-smooth etremum! A Which path does the light ray take then? B Glass Air 24

25 Refraction of light; non-smooth solution l g L l a A H B h Glass s g Air s a = speed of light ray changes at the interface between the two media. We do not know for what value, the bend takes place. This is given by variable end conditions. Let us see 25

26 Intermediate variable end condition l g L l a A H h Glass s a y s g Now, for the two parts, c is a variable end condition! B Air 26

27 Broken etremal conditions for a light ray l g L l a A H h Glass s a y s g B Air 27

28 Snell s law from the corner condition yf y = v 1 1+ y 2 is continuous at the corner. So, The first corner condition also holds good here. Because is zero. Thus, we derived Snell s law using calculus of variations. 28

29 The end note Variable end conditions General variation; transversality conditions and broekn etremals General variation Variable end conditions for first and second derivatives cases Transversality conditions Transversality conditions for the two-function case Broken etremals Weierstrass-Erdmann corner conditions Thanks 29