Variation and Extremum of a Functional

Consider a set of admissible functions on which a functional is defined. If one of the admissibility conditions is u = 0 at x = 0, then a typical function from this set can be represented as shown in Fig. 2.3(a).

Figure 2.3 (a)	Figure 2.3 (b)

A small change in the argument of I is called as variation (or first variation) of u and is denoted by u. It is defined as a small change in the value of u at every point of the interval [0, L]. Note that u is a function of x over the interval [0, L]. Since u is constrained at x = 0, no change is expected in u at x = 0. Therefore, the variation u takes the value zero at the end-point of the interval where u is specified. The function u is shown in Fig. 2.3(b). Since the value of u is small everywhere, it is often expressed as

u =

(2.17)

where

= a small number and

= an arbitrary function of x which satisfies the condition = 0 at x = 0.

This small change in the argument of I induces a change in the value of I. It is called as the variation (or first variation) of I and is denoted by . To find the expressions of , we consider the value of the functional at the argument or . Note that I( ) is a function of for given u and . Expanding about = 0, we get

(2.18)

The quantity represents the variation . Thus,

= =

(2.19)

The functional I is said to be extremum (minimum or maximum) when or is zero. The corresponding extremizing function u is found from the condition

(2.20)

for every function such that = 0 at x = 0 and belongs to the set of admissible functions.

While deriving the variational integral form from equation (2.6), some properties of the variational operator are needed. They are stated next. Note that is an operator similar to the differential operator d operating on usual functions. So, the operator obeys certain properties of d operator like the product rule etc. Further, operator commutes with d operator, integral operator etc. the relevant properties needed in the derivation are:

(i)	,	(2.21)
(ii)	,      (f is known function or a constant),	(2.22)
(iii)	,	(2.23)
(iv)	,	(2.24)
(v)	,	(2.25)
(vi)	,	(2.26)
(vii)		(2.27)