Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx
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1 Tutoril 2 Euler Lgrnge In one entene: d Fy = F d Importnt ft: ) The olution of EL eqution i lled eterml. 2) Minmum / Mimum of the "Mot Simple prolem" i lo n eterml. 3) It i eier to olve EL nd hek if we reeived olution for our prolem. 4) EL i tully eond order prtil eqution. Mot of the prolem do not hve n ey olution. We olve them numerilly. 5) In the theory of Clulu of Vrition there re more term tht del with: Mny vrile. Higher derivtive. Suffiient ondition for minimum / mimum Corner point Multi dimenionl Integrl In the following lide we will prove EL eqution in it mot imple vrition. We will how reult for min/m deiion nd 2D integrl. The proof will rely on 3 lemm. Lemm u() i ontinue funtion on []. [ ] h = Suppoe tht Ih= u( h ) ( d ) = C[ ] h = h Then u()= on [] Proof Lemm Let ume tht eit in [] o tht u( ). Wlog u( ) >. Sine u i ontinue tht eit [''] uh tht u( ) > ('') Then we will define h(): ( ' ) 2 ( ' ) 2 [ ' ' ] h = otherwie Then h fulfill our demnd given ove. But: ' ' u h d> in ontrdition!
2 Lemm 2 (Devo Rmon) Let v( ) C [ ] nd let ume ' = h C [ ] = = ' [ ] v h d h h h C for ll ontinue point Then eit ontnt tht v =. Proof Lemm 2 We will define v( ) Then note tht = d ( v ) d= v( ) d ( ) = Now we will define Note tht h()=h()= h' = v [ ] h C Let lulte: h = v t dt 2 ( v( ) ) d= ( v( ) ) h' ( ) d= v( ) h' ( ) h' ( ) d= Then v()= Remrk: The lultion ove hould e mde for eh ontinue prt ut the onluion i the me. Lemm 3 Given u( ) C [ ] v( ) C [ ] We ume: [ ] ( ' ) [ ] I h = u h + v h d= h C h = h = Then: v C [ ] v = u t dt+
3 Proof lemm 3 Let U( ) = u t dt Then U '( ) = u( ) u h d U h d U h U h = ' = ' So ' [ ] ' I y = u h + v h d= U + v h d From lemm 2 we get: U + v = v = u t dt+ Reminder From Tutoril : We would like to meure ditne etween the funtion in order to e le to define n etermum: f f = up [ ] X [ ] f ( ) = up [ ] X [ ] f ( ) C C f = f + f ' [ ] [ ] [ ] C C C f = f + f ' [ ] [ ] [ ] C C C Now we n define minimum for the funtionl: I y e funtionl in Σ C Let [ ] [ ] Wek Lol Minimum y ( ) i wek lol minimum iff ε < ε [ ] [ ] > y y-y I y I y C [ ] Strong Lol Minimum y ( ) i wek lol minimum iff ε < ε > y y-y I y I y Quetion: [ ] [ ] [ ] C Strong Wek or Wek Strong? Prove it!!!
4 The firt Vrition In funtion theory we lerned tht the funtion h n etermum when the firt derivtive i ero. In imilr wy we will define the firt vrition nd prove tht if the funtionl reh n etermum it firt vrition mut e ero. If y C [ ] We will y tht h( ) C [ ] elong to ε t < ε y + th D y of "llowed vrition" iff In thoe ondition the funtion: ϕ () t = I[ y + th] i defined in the neighorhood of t= y h Definition: di [ y + th] ϕ ' y h = i lled the firt vrition of y in the diretion of h. dt t= We will define: d δ I[ h] = I[ y + th] dt t= Theorem y i wek lol minimum of the funtionl I[ y] y Σ If And elong to n e derived round. Then δ I[ h] = h( ) D( y ) of "llowed vrition" nd the funtion ϕ () t = I[ y + th] Proof of Theorem (*) [ ] [ ] y-y C [ ] I y I y < ε ε So if t < then the ondition re vlid h [ ] Sine h( ) elong to D( y ) of "llowed vrition" there i δ uh tht t < δ y + th Σ nd there i ugroup tht will lo mke (*) vlid. Sine: ϕ = I y ϕ t = I y +th [ ] () [ ] Then t= i lol minimum of ϕ. Aording to funtion theory ϕ ' =.
5 Reminder From Tutoril (The mot imple prolem) [ ] I y = F y y' d min = = y A y B y C [ ] Quetion: { y C y A y B} Our domin i [ ]: If we hooe y Σ then wht i Σ= = = Solution y = y + th Σ y = A y( A) y = B y = B h = h = [ ] { : } D y = h C h = h = Theorem 2 F y C R D y of llowed vrition? 3 [ ] [ ] = { y( ' ) + ( ' ) '} y h C δ I h F y y h F y y h d Proof of theorem 2 The proof given hould e given to eh ontinue prt. Due to linerity in finite um integrl the full proof i imilr. ( ) [ ] = + ' + ' I y F y th y th d Aording to lynih: = { y( + + ) + Z ( + + ) } I ' t F y thy ' th' h F y thy ' th' h' d So for t= we get the deired funtion.
6 Euler Lgrnge Theorem Integrl verion Aume: [ ] I y = F y y' d min 3 = = [ ] y A y B y C F C R Then: Eit ontnt uh tht: ( ' ) y () ' () F y y = F t y t y t dt+ (mye not in the non ontinue point) Proof We will ue theorem 2 nd lemm 3. From theorem δ I[ h] = From theorem + 2 (*) [ ] { y } From (*) nd lemm 3: There i ontnt tht δ I h = F h+ Fh' d= ( ' ) y () ' () F y y = F t y t y t dt+ (mye not in the non ontinue point)
7 Euler Lgrnge Theorem y ' i not ontinue ut in eh ontinue prt we will ue Newton-lynih. We n write: d F y y F t y t y t d ( ' ) = y () ' () Or in hort: d F = F d y
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