Calculus of Variations Basics

Transcription

1 Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y 2,,y n ), 1.1) begnnng wth the case where 1.1) depends on a sngle functony and hence reduces to J [y] F x,y,y ). 1.2) We assume that all admssble curves are smooth 1, but, the end ponts of the curves for whch 1.2) s defned can move n an arbtrary way. Now, we also defne the dstance between two curvesy yx) and y y x) as ρy,y ) max y y +maxy y +ρp0,p0 )+ρp 1,P1 ), 1.3) wherep 0 andp 0 denote the left-hand end ponts of the curvesy andy, respectvely, andp 1 and P 1 denote the rght-hand end ponts. In general, the functons y and y are defned on dfferent ntervalsi and I. Thus, n order for 1.3) to make sense, we have to extendy and y onto some nterval contanng bothi andi e.g., regon[,x 1 +δx 1 ] as shown n Fg. 1.1). Now, let y and y be two neghborng curves, n the sense of dstance 1.3), and let hx) y y. Moreover, let P 0,y 0 ) andp 1 x 1,y 1 ) 1 We say the functon yx) s smooth n an nterval [,x 1 ], f both the functon as well as t s dervatves are contnuous n[,x 1 ]. 1

2 Fgure 1.1: Functonsy andy and ther correspondng domans denote the end ponts of the curvey, whle the end ponts of the curvey are denoted by P 0 +δ,y 0 +δy 0 ) andp 1 x 1 +δx 1,y 1 +δy 1 ). The correspondng varaton δj of the functonal J [y] s defned as the expresson whch s lnear nh,h,δ,δy 0,δx 1,δy 1, and dffers from the ncrement J J [y +h] J [y] by a quantty of the order hgher than1relatve toρy,y+h). Snce J +δx 1 +δ F x,y +h,y +h ) + [F x,y +h,y +h ) F x,y,y )] +δx 1 x 1 F x,y +h,y +h ) F x,y,y ) x0 +δ F x,y +h,y +h ), t follows by usng Taylor s theorem and lettng the symbol denote equalty except for terms of order hgher than1relatve toρy,y+h) that J [F y x,y,y )h x,y,y )h ] + F x,y,y ) xx1 δx 1 F x,y,y ) xx0 δ [ F y d ] hx)+ ) h xx1 h xx0 + F xx1 δx 1 F xx0 δ. The term nsde the round brackets of the above equaton s due to ntegraton by parts. However, t s clear from Fg.1.1 that h ) δy 0 y )δ, hx 1 ) δy 1 y x 1 )δx 1, 2

3 where has the same meanng as before, and hence [ δj F y d ] hx)+ δy xx 1 x + F y )δx xx 1 x, where we defne δx xx δx and, δy xx δy 0,1). Next, we return to the more general functonal 1.1), whch depends onnfunctonsy 1,y 2,,y n. From the analogy, varaton for the general functonal s gven as δj F y d ] } ) xx F y h x) + δy xx F F xx y 0 y δx 1.4) where, as before, we defne δx xxj δx j and, δy xx δy j j 0,1). x 1.2 Varaton n terms of the Canoncal Varables We now wrte an even more concse formula for the varaton 1.4), at the same tme ntroducng some new mportant deas. Let and suppose that the Jacoban p 1,,n), p 1,,p n ) y 1,,y n ) det Fy y k s nonzero so that the mappng s nvertble). Then we can solve the equatons fory 1,,y n as functons of the varables x,y 1,,y n,p 1,,p n. 1.5) Next we can express the functonf x,y 1,y 2,,y n,y 1,y 2,,y n) n terms of a new functon Hx,y 1,,y n,p 1,,p n ) related tof by the formula H F + y F + p y, 1.6) where y are regarded as the functons of the varables. The functon H s called the Hamltonan functon) correspondng to the functonal J [y 1,,y n ]. In ths way, we can make a local transformaton from the varables x,y 1,y 2,,y n,y 1,y 2,,y n,f to the new quanttes x,y 1,,y n,p 1,,p n,h called the canoncal varables. In terms of the canoncal varables, varaton can be wrtten n the form δj F y dp ] } h x) + p δy Hδx) xx 1 x. 1.7) 3

4 1.3 Euler-Lagrange Equatons & Correspondng Constrants The necessary condton for the dfferentable functonal J [y 1,y 2,,y n ] to have an extremum for[ŷ 1,ŷ 2,,ŷ n ] s that ts varaton vansh for[ŷ 1,ŷ 2,,ŷ n ],.e., that δj [h] 0 for[ŷ 1,ŷ 2,,ŷ n ] and all admssbleh. So, from 1.4) and 1.7), the necessary condtonδj [h] 0 mples F y d ] } F y h x) + δy xx 1 x + F y dp ] } h x) + F ) y δx xx 1 x p δy Hδx) xx 1 x ) Suppose the functonal J [y 1,y 2,,y n ] has an extremum n certan class of admssble curves) for some curve y ŷ jonng the ponts P 0,y 0 1,,y0 n) andp1,y 1 1,,y1 n). Then, snce J [y 1,y 2,,y n ] has an extremum for ŷ compared to all admssble curves, t certanly has an extremum forŷ compared to all curves wth fxed 2 end pontsp 0 andp 1. Therefore, F y d ] } F y h x) Because of the ndependence, the above equaton smply reduces to F y dp ] } h x) 0. F y d ] } F y h x) F y dp ] } h x) 0 1.9) for allvalues. Lemma 1. If αx) s contnuous n[a,b], and f b a αx)hx) 0 for every contnuous functonhx) such thatha) hb) 0, thenαx) 0 for allxn[a,b]. 2 When we say fxed, t obvously meansh ) h ) 0,δx j 0 andδy j 0. 4

5 So, from the above lemma and 1.9), F y d ) F y F y dp ) 0, 1,,n). 1.10) The above equatons known as Euler-Lagrange equatons represent a system of n second-order 3 dfferental equatons and ts soluton can be obtaned from 2n arbtrary constrants gven as ) δy + F F xx y 0 y δx xx0 p δy xx0 Hδx xx0 0, 1,,n), 1.11) δy + F xx 1 ) y δx xx1 p δy xx1 Hδx xx1 0, 1,,n). 1.12) The above equaton s from 1.8) and due to the ndependence ofδ andδx 1. Before concludng, a few mportant relatons between F and H are gven below. These relatons can be derved from 1.6). H x F x, H y F y and H p y, 1,,n) 1.13) Substtutng the above relatons nto 1.10) gves2n frst order dfferental equatons dy H p and dp H y, 1,,n). 1.14) Once agan, 2n arbtrary constrants requred to solve the above equatons are gven by 1.11) and 1.12). The materal presented here s from the book: [1] Gelfand, I. M., Fomn, S. V., Calculus of varatons, Prentce-Hall, Inc., or2n frst order dfferental equatons n the case of canoncal varables, whch are gven by 1.14). 5