knew to get stationary point which minimizers is given functional ,we consider associated x 8) = do + a, (t - to ) + a, (t- to) (t-ti) + as(t- to) (t- to) (t-to) Euler - Lagrange's equation : = CX , 0 + Cx,I 't + (x," t" + Cast similarly , yeb) = Cyno + Cyil it + Cyis th + Cyist ? OF - ( OF ) + ( 21 ) = 0 3 ( t) = (yoo + Col . t + (it " + gist 7 of ( 2W " ) = 0 =0 ( integrating wir. to it ) AL = A. for t = to , ( again integrating ) X. = X(t.) = (x,o + Cx,1" to + Canto + Cris to 7 he = Aot + Al 30 = y (to ) = Cyno + Gil'to + Cyia to + Cyis to3 ( again integrating ) 7 dre = Act? + Ait + Az 30 = 3( t. ) = Cyno + Caprito + Gret. 2 + Cqis tos (again interesting ) * web = Act3 + Alt? + Art + A3 Let co = A3 , C1 = Az , Cz= Al and C3 = to where Ao, Al, As, As are for teti (ie. 1, 2,3 ) in general, arbitrary constants . Xi = Cx/o + Canti + Cx/2 ti + Cxiste So we get web = Cot cit test ? + est 3 Hi = Cyro + Cgil ti + Cuizte + Cyis ti 3i = Gre + Grote + Grati + gas ti - W( to ) = Wo # Co + cito + Cator + Cyto = W. wti) = WI # Co + Gti + cati + esti = w, with) = W2 7 Co + cite thats tests = Wz Webs ) = W3 # Co tats + cats' + Cat? = Ws For x , Cx 1 0 for y , yo Cy ro V Cx, 1 = VX, Cy, 1 = VI In matrix form , Cuzz = VX 2 wo 4/3 Cx 1 3 Cy , 3 - - W , (310 W2 For 31 W3 E V egel = VX3 VC = W where V z to C = Co and W = wo WI (x 10 Cyro XO yo W2 ve = V Call Cy , ' = [Vx Vx2 VXy] X . W3. C X , 2 Cy.3 (3.3. w Thus , we get the stationary path is given functional is web) = Co + cit + Cat? + Cit3 where Co , C , CziCs obtained from matrix equation VC = W.Let to