First Variation of a Functional

Transcription

1 First Variation of a Functional The derivative of a function being zero is a necessary condition for the etremum of that function in ordinary calculus. Let us now consider the equivalent of a derivative for functionals because it plays the same crucial role in calculus of variations as does the derivative of the ordinary calculus in minimization of functions. Let us begin with a simple but a very important concept called a Gâteau variation. Gâteau variation The functional δ J( ) is called the Gâteau variation of J at when the limit that is defined as follows eists. IISc 1 Ananthasuresh

2 ( + ε ) ( ) δ J( h ; ) = lim J h J where h is any vector in a vector space, X. ε 0 ε Let us look at the meaning of h and ε geometrically. Note that h, X. Now, since is the unknown function to be found so as to minimize (or maimize) a functional, we want to see what happens to the functional J( ) when we perturb this function slightly. For this, we take another function h and multiply it by a small number ε. We add ε h to and look at the value of J( + εh). That is, we look at the perturbed value of the functional due to perturbation ε h. Symbolically, this is the shaded area shown in Fig. 1 where the function is indicated by a thick solid line, h by a thin solid line, and + εh by a thick dashed line. Net, we think of the situation of ε tending to zero. As ε 0, we consider the limit of the shaded area divided by ε. If this limit eists, such a limit is called the IISc 2 Ananthasuresh

3 Gâteau variation of J( ) at for an arbitrary but fied vector h. Note that, we denote it as δ J( h ; ) by including h in defining Gâteau variation. +εh Function h Domain D(X) Figure 1. Pictorial depiction of variation ε h of a function IISc 3 Ananthasuresh

4 Although the most important developments in calculus of variations happened in 17 th and 18 th centuries, this formalistic concept of variation was put forth by a French mathematician Gâteau around the time of the First World War. So, one can say that intuitive and creative thinking leads to new developments and rigorous thinking makes them mathematically sound and completely unambiguous. To reinforce our understanding of the Gâteau variation, let us relate it to the concept of a directional derivative in multi-variable calculus. A directional derivative of the function ( 1, 2,..., n ) compact form as ( ) ε 0 h f ( + ε ) ( ) lim f h f ε f denoted in a, in the direction of a unit vector h is given by. IISc 4 Ananthasuresh

5 Here the vector is the usual notion that you know and not the etended notion of a vector in a vector space. We are using the over-bar to indicate that the denoted quantity consists of several elements in an array as in a column (or row) vector. You know how to take the derivative of a f with respect to any of its variables, say,1 i n. It is function ( ) simply a partial derivative of f ( ) with respect to i IISc 5 Ananthasuresh i. You also know that this partial derivative indicates the rate of change of f ( ) in the direction. What if you want to know the rate of change of f ( ) in some of i arbitrary direction denoted by h? This is eactly what a directional derivative gives. Indeed, f ( ) = f ( ) h = f ( ) T h. That is, the h h h component of the gradient in the direction of h. Now, relate the concept of the directional derivative to Gâteau variation because we want to know how the value of the functional changes in a

6 direction of another element h in the vector space. Thus, the Gateau variation etends the concept of the directional derivative of finite multivariable calculus to infinite dimensional vector spaces, i.e., calculus of functionals. Gâteau differentiability If Gateau variation eists for all h X then J is said to be Gateau differentiable. Operationally useful definition of Gâteau variation Gateau variation can be thought of as the following ordinary derivative evaluated at 0 ε =. IISc 6 Ananthasuresh

7 d δj h J h d ( ; ) = ( + ε ) ε ε = 0 This helps calculate the Gâteau variation easily by taking an ordinary derivative instead of evaluating the limit as in the earlier formal definition. Note that this definition follows from the earlier definition and the concept of how an ordinary derivative is defined in ordinary calculus if we think of the functional as a simple function of ε. Gâteau variation and the necessary condition for minimization of a functional Gâteau variation provides a necessary condition for a minimum of a functional. IISc 7 Ananthasuresh

8 Consider ( ) J where J( ), D, is an open subset of a normed vector * space X, and D and any fied vector h X. If * is a minimum, then * * ( ε ) ( ) J + h J 0 must hold for all sufficiently small ε Now, IISc 8 Ananthasuresh

9 for ε 0 and for ε 0 * * ( + ε ) ( ) J h J ε * * ( + ε ) ( ) J h J ε 0 0 This simple derivation proves that the Gâteau variation being zero is the necessary condition for the minimum of a functional. Likewise we can IISc 9 Ananthasuresh

10 show (by simply reversing the inequality signs in the above derivation) that the same necessary condition applies to maimum of a functional. Now, we can state this as a theorem since it is a very important result. Theorem: necessary condition for a minimum of a functional ( ) δ J * ; h = 0 for all h X Based on the foregoing, we note that Gâteau variation is very useful in the minimization of a functional but the eistence of Gateau variation is a weak requirement on a functional since this variation does not use a norm in X. Without a norm, we cannot talk about continuity of a IISc 10 Ananthasuresh

11 functional because we cannot judge how close two functions are to each other. Thus, Gâteau variation is not directly related to the continuity of a functional. For this purpose, another differential called Fréchet differential has been put forth. Frechet differential ( + ) ( ) ( ; ) J h J dj h lim h 0 h = 0 If the above condition holds and ( ) ; dj h is a linear, continuous functional of h, then J is said to be Fréchet differentiable at with increment h. IISc 11 Ananthasuresh

12 dj ( ; h ) is called the Fréchet differential. If J is differentiable at each D we say that J is Fréchet differentiable in D. Some properties of Fréchet differential i) ( ) ( ) ( ; ) ( ; ) J + h = J + dj h + E h h for any small non-zero h X has a limit zero at the zero vector in X. That is, ( ) lim E h ; = 0. h θ IISc 12 Ananthasuresh

13 Based on this, sometimes the Fréchet differential is also defined as follows. lim h θ ( + ) ( ) ( ; ) J h J dj h h = 0. ii) ( ) ( ) dj ; a1h1+ a2h2 = a1dj ; h1 + a2dj ( ; h2) must hold for any numbers a1, a2 K and any h 1, h 2 X. This is simply the linearity requirement on the Fréchet differential. iii) ( ) dj ; h c h for all h X, where c is a constant. This is the continuity requirement on the Fréchet differential. IISc 13 Ananthasuresh

14 iv) This is to say that the Fréchet differential is a linear functional of h. Note that it also introduces a new definition: Fréchet derivative, which is simply the coefficient of h in the Fréchet differential. Relationship between Gâteau variation and Fréchet differential If a functional J is Fréchet differentiable at then the Gateau variation of J at eists and is equal to the Fréchet differential. That is, ( ) ( ) δ J h ; = dj h ; for all h X IISc 14 Ananthasuresh

15 Here is why: Due to the linearity property of dj ( ; h ), we can write ( ; ε ) = ε ( ; ) dj h dj h By substituting the above result into property (i) of the Fréchet differential noted earlier, we get ( ) ( ) ( ) ( ) J + εh J εdj ; h = E, εh h ε for any h X A small rearrangement of terms yields IISc 15 Ananthasuresh

16 ( + ε ) ( ) J h J ε ( ; ) (, ε ) = dj h + E h h ε ε When limit ε 0 is taken, the above equation gives what we need to prove: ( + ε ) ( ) J h J ε lim = δj( h ; ) = dj( h ; ) because lim E(, εh) h = 0 ε 0 ε ε 0 ε Note that the latter part of property (i) is once again used in the preceding equation. Operations using Gateau variation Consider a simple general functional of the form shown below. IISc 16 Ananthasuresh

17 2 ( ) ( ) =, ( ), ( ) where J y F y y d 1 ( ) y = dy d Note our sudden change of using. It is no longer a member (element, vector) of a normed vector space X. It is now an independent variable and defines the domain of y, ( ) which is a member of a normed vector space. Now, y ( ) is the unknown function using which the functional is defined. We need to have our wits about us to see which symbol is used in what way! If we want to calculate the Gâteau variation of the above functional, instead of using the formal definition that needs an evaluation of the limit IISc 17 Ananthasuresh

18 we should use the alternate operationally useful definition taking the J y+ εh with respect to ε and evaluating at ε = 0. ordinary derivative of ( ) In fact, there is an easier route that is almost like a thumb-rule. Let us find that by using the derivative approach for the above simple functional. 2 ( ) ( + ε ) =, ( ) + ε ( ), ( ) + ε ( ) J y h F y h y h d 1 Recalling that δj( ; h) J( εh) d = +, we can write d ε ε = 0 IISc 18 Ananthasuresh

19 2 d d J ( + εh) = F (, y+ εh, y εh ) d dε dε d = { F (, y + ε h, y + ε h )} d dε 1 Please note that the order of differentiation and integration have been switched above. It is a legitimate operation. By using chain-rule of differentiation for the integrand of the above functional, we can further simplify it to obtain 2 2 F F F F δ J ( ; h) = h + h = h h d ( y εh) ( y εh + ) y y ε = 0 IISc 19 Ananthasuresh

20 What we have obtained above is a general result in that for any functional, be it of the form Jyy (,,, y, y, ), we can write the variation as follows. 2 2 F F F F δ J ( ; h) = F(, y, y, y, y, ) d = h + h + h + h + d y y y y. 1 1 Note that in taking partial derivatives with respect to y and its derivatives we treat them as independent. It is a thumb-rule that enables us to write the variation rather easily by inspection and using rules of partial differentiation of ordinary calculus. We have now laid the necessary mathematical foundation for deriving the Euler-Lagrange equations that are the necessary conditions for the etremum of a function. Note that the Gâteau variation still has an IISc 20 Ananthasuresh

21 arbitrary function h. When we get rid of this, we get the Euler-Lagrange equations. For that we need to talk about fundamental lemmas of calculus of variations. Variational Derivative We have studied Gâteau variation and Fréchet differential and the relationship between them. There is one more subtle variant of this, which is called the variational derivative. It is useful in some applications and in proving some theorems. More importantly, it tells us an alternative way of looking at the concept of variation based purely on the techniques of ordinary calculus. In fact, it can be interpreted as the partial derivative equivalent for calculus of variations. As the history goes, Euler had apparently derived his eponymous necessary condition using this concept. IISc 21 Ananthasuresh

22 Let us begin with the notation. The variational derivative of a functional J f = F(, y, y ) d is denoted as 0 δ J δ y and is given by δ J d F = Fy δ y d y (1) You may observe that it is nothing but the Euler-Lagrange epression that δ should be zero. When J has a more general form, the epression for J δ y will be the corresponding epression in the E-L equation that we equate to zero. Let us see what rationale underlies this definition. Because we want to use only the techniques of ordinary calculus, let us discretize y ( ) and consider finitely many discrete points k 1,2,..., N. See Fig. 1. As can be seen in k ( ) = within the interval (, 0 f ) IISc 22 Ananthasuresh

23 this figure, by way of discretization, we are approimating the continuous curve of y ( ) by a polygon. y ( ) σk = δy Figure 1. Discretization of a continuous curve y ( ) by a polygon. All subdivisions on the -ais are equal to. A local perturbation at k is considered and its effect is shown with the dashed lines. k N f IISc 23 Ananthasuresh

24 Now, the functional can be approimated as follows. N N N ( yk+ 1 y ) k ( yk+ 1 yk) J JN = F k, yk, ( k+ 1 k) = F k, yk, k= 1 k= 1 ( k+ 1 k) k= 1 (2) where in the last step we have assumed that all subdivisions along the - ais are equal to. Our variables to minimize J N are now { y1, y2,, yn }. Consider the partial derivative of J N with respect to y k. Here, we have just used the chain rule of differentiation. As 0, the RHS of Eq. (3) goes to zero. Now, divide the LHS and RHS of Eq. (3) by to get (3) IISc 24 Ananthasuresh

25 ( y y ) ( y y ) F y F y JN ( yk+ 1 yk) = Fy( k, yk, ) + yk (4) k k 1 k+ 1 k y ( k 1, k 1, ) y ( k, k, ) When 0, yk, which can be interpreted as the shaded area in Fig. 1, also tends to zero. In fact, we then denote y k as σ k or, in general, simply as δ y evaluated at = k. Furthermore, as 0, JN J. We take the limit of Eq. (4) as 0. lim 0 J N δ J d = = Fy y δ y d k ( Fy ) (5) IISc 25 Ananthasuresh

26 Notice how we defined the variational derivative in Eq. (5). We can think δ J J( y+ h) J( y) of as the limiting case of where h is the perturbation δ y σ (i.e., variation) of y at some and σ is the etra area under y ( ) due to that perturbation. Therefore, we write δ J J = J( y+ h) J( y) = + ε σ δ y = (6) where ε is a small discretization error. When the discretization error is insignificantly small, we can write δ J J σ (7) δ y = IISc 26 Ananthasuresh

27 Thus, the variational derivative helps us get the first order change in the value of the functional for a local perturbation of y ( ) at = ˆ. Think of Taylor series of epansion of a function of many variables and try to relate this concept of first order change in the value of the function. IISc 27 Ananthasuresh