II. First variation of functionals

Transcription

1 II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent of a erivative for functionals because it plays the same crucial role in calculus of variations as oes the erivative of the orinary calculus in minimization of functions. Let us begin with a simple but a very important concept calle a Gâteau variation. Gâteau variation The functional δ J ( ) is calle the Gâteau variation of J at when the limit that is efine as follows eists. δ J ( ; h ) = lim J h J where h is any vector in a vector space,. Let us look at the meaning of h an geometrically. Note that, h. Now, since is the unknown function to be foun 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 an multiply it by a small number. We a h to an look at the value of J ( + h). That is, we look at the perturbe value of the functional ue to perturbation h. This is the shae area shown in Fig. 1 where the function is inicate by a thick soli line, h by a thin soli line, an + h by a thick ashe line. Net, we think of the situation of tening to zero. As 0, we consier the limit of the shae area ivie by. If this limit eists, such a limit is calle the Gâteau variation of J ( ) at for an arbitrary but fie vector h. Note that, we enote it as δ J ( h ; ) by incluing h in efining Gâteau variation. Function +h h Domain D() Figure 1. Pictorial epiction of variation h of a function Although the most important evelopments in calculus of variations happene in 17 th an 18 th centuries, this formalistic concept of variation was put forth by a French mathematician Gâteau aroun the time of the first worl war. So, one can say that intuitive an creative thinking leas to new evelopments an rigorous thinking makes them mathematically soun 1 of 6 Ananthasuresh, IISc

2 an completely unambiguous. To reinforce our unerstaning of the Gâteau variation efine as above, let us relate it to the concept of a irectional erivative in multi-variable calculus. A irectional erivative of the function ( 1, 2,..., n ) f ( ) in the irection of a given vector h is given by f enote in a compact form as lim f h f. Here the vector is the usual notion that you know an not the etene notion of a vector in a vector space. We are using the over-bar to inicate that the enote quantity consists of several elements in an array as in a column (or row) vector. You know how to take the erivative of a function f ( ) with respect to any of its variables, say i, 1 i n. It is simply a partial erivative of f ( ) with respect to i. You also know that this partial erivative inicates the rate of change of f ( ) in the irection of i. What if you want to know the rate of change of f ( ) in some arbitrary irection enote by h? This is eactly what a irectional erivative gives. Now, relate the concept of the irectional erivative to Gâteau variation because we want to know how the value of the functional changes in a irection of another element h in the vector space. Thus, the Gateau variation etens the concept of the irectional erivative of finite multi-variable calculus to infinite imensional vector spaces, i.e., calculus of functionals. Gâteau ifferentiability If Gateau variation eists for all h then J is sai to be Gateau ifferentiable. Operationally useful efinition of Gâteau variation Gateau variation can be thought of as the following orinary erivative evaluate at = 0. δj h J h ( ; ) = ( + ) = 0 This helps calculate the Gâteau variation easily by taking an orinary erivative instea of evaluating the limit as in the earlier formal efinition. Note that this efinition follows from the earlier efinition an the concept of how an orinary erivative is efine in orinary calculus if we think of the functional as a simple function of. Gâteau variation an the necessary conition for minimization of a functional Gâteau variation provies a necessary conition for a minimum of a functional. Consier where J ( ), D, is an open subset of a norme vector space an an any fie vector h If * is a minimum, then * D 2 of 6 Ananthasuresh, IISc

3 ( ) ( ) J + h J 0 must hol for all sufficiently small Now, for 0 J( + h) J( ) 0 an for 0 J( + h) J( ) 0 If we let 0, J( + h) J( ) lim 0 0 > J h J 0 lim J( + h) J( ) an lim 0 0 < eistence of Gâteau variation ensures the eistence of this limit = δ J( ; h) = 0 This simple erivation proves that the Gâteau variation being zero is the necessary conition for the minimum of a functional. Likewise we can show (by simply reversing the inequality signs in the above erivation) that the same necessary conition applies to maimum of a functional. Now, we can state this as a theorem since it is a very important result. Theorem: necessary conition for a minimum of a functional ( ) δ J * ; h = 0 for all h Base on the foregoing, we note that the 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 oes not use a norm in. Thus, it is not irectly relate to the continuity of a functional. For this purpose, another ifferential calle Fréchet ifferential has been put forth. Frechet ifferential lim h 0 ( + ) ( ) ( ; ) J h J J h h = 0 If the above conition hols an ( ) ; J h is a linear, continuous functional of h, then J is sai to be Fréchet ifferentiable at with increment h. 3 of 6 Ananthasuresh, IISc

4 J ( ; h ) is calle the Fréchet ifferential. If J is ifferentiable at each D we say that J is Fréchet ifferentiable in D. Some properties of Fréchet ifferential i) ( ) ( ) ( ; ) ( ; ) J + h = J + J h + E h h for any small non-zero h has a limit zero at the zero vector in. That is, h 0 in ( ) lim E ; h = 0. Base on this, sometimes the Fréchet ifferential is also efine as follows. h 0 ( + ) ( ) ( ; ) J h J J h lim h ii) ( ) ( ) = 0. J ; a1h1+ a2h2 = a1j ; h1 + a2j ( ; h2) must hol for any numbers a 1, a 2 K an any h1, h2. This is simply the linearity requirement on the Fréchet ifferential. iii) ( ) J ; h constant h for all h This is the continuity requirement on the Fréchet ifferential. iv) ( ; ) = ( ) J h J h { Frechet erivative This is to say that the Fréchet ifferential is a linear functional of h. Note that it also introuces a new efinition: Fréchet erivative, which is simply the coefficient of h in the Fréchet ifferential. Relationship between Gâteau variation an Fréchet ifferential If a functional J is Fréchet ifferentiable at then the Gateau variation of J at eists an is equal to the Fréchet ifferential. That is, Here is why: ( ) ( ) δ J h ; = J h ; for all h Due to the linearity property of J ( ; h ), we can write ( ; ) = ( ; ) J h J h 4 of 6 Ananthasuresh, IISc

5 Substituting the above result into property (i) of the Fréchet ifferential note earlier, we get ( ) ( ) ( ) ( ) J + h J J ; h = E, h h for any h A small rearrangement of terms yiels J h J = J ( ; h) + E (, h) h When limit 0 is taken, the above equation gives what we nee to prove: J h J lim = δj( ; h) = J( ; h) because lim E(, h) h = 0 Note that the latter part of property (i) is once again use above. Operations using Gateau variation Consier a simple general functional of the form shown below. 2 ( ) = (, ( ), ( )) J y F y y 1 y where y ( ) = Note our suen change of using. It is no longer a member (element, vector) of a norme vector space. It is now an inepenent variable an efines the omain of y, ( ) which is a member of a norme vector space. Now, y ( ) is the unknown function using which the functional is efine. We nee to have our wits about you to see which symbol is use in what way! If we want to calculate the Gâteau variation of the above functional, instea of using the formal efinition that nees an evaluation of the limit we shoul use the alternate J y+ h with respect to operationally useful efinition taking the orinary erivative of ( ) an evaluating at = 0. In fact, there is an easier route that is almost like a thumb-rule. Let us fin that by using the erivative approach for the above simple functional. 2 ( + ) = (, ( ) + ( ), ( ) + ( )) J y h F y h y h 1 Recalling that δj( ; h) J( h) = +, we can write = 0 5 of 6 Ananthasuresh, IISc

6 2 J ( + h) = F(, y+ h, y h ) = { F (, y + h, y + h )} 1 Please note that the orer of ifferentiation an integration have been switche above. It is a legitimate operation. By using chain-rule of ifferentiation for the integran of the above functional, we can further simplify it to obtain 2 2 F F F F δ J ( h ; ) = h h h h + = + ( y h) ( y h ) + + y y. 1 = 0 1 What we have obtaine above is a general result in that for any functional, be it of the form J(, y, y, y, y L, ), we can write the variation as follows. 2 2 F F F F δ J ( ; h) = F(, y, y, y, y, L) = h + h + h + h + L. y y y y 1 1 Note that in taking partial erivatives with respect to y an its erivatives we treat them as inepenent. It is a thumb-rule that enables us to write the variation rather easily by inspection an using rules of partial ifferentiation of orinary calculus. We have now lai the necessary mathematical founation for eriving the Euler-Lagrange equations that are the necessary conitions for the etremum of a function. Note that the Gâteau variation still has an arbitrary function h. When we get ri of this, we get the Euler- Lagrange equations. For that we nee to talk about funamental lemmas of calculus of variations. 6 of 6 Ananthasuresh, IISc