By Jurgen Jost, Xianqing Li-Jost

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Proof. t. s at s = 0 to obtain ~ (Fpp(t, u(t), u(t))fj(t) + Fpu(t, u(t), ii(t))rj(t)) -Fpu(t, u(t),u(t))fj(t) - Fuu{t, u(t), ii(t))rf(t) = 0. e. 11). d. 2. Let a < a\ < a2 ** is homogeneous of second order in (77,71-), we have 20(t, 77, IT) = (pr,(t, ry, TT)77 4- • Therefore r«2 2/ **

v(t,V,v)-V + 4>At,il,v)-v}dt.

LetF e C 3 ( J x R d x R d , R ) and suppose ue C2{I,Rd). Suppose that Fpp(t,u(t),ii(t)) is positive definite on I. If there exists a* with a < a* < b that is conjugate to a, then u cannot be a local minimum of I. More precisely, for any e > 0, there exists v E Dl(I, Rd) with v(a) = u(a), v(b) = u(b), sup (\u(t) - v(t)\ + \ii(t) - v±(t)\) < e tei and I(v) < I(u). Proof. Let rj(t) be a nontrivial Jacobi field on [a, a*]. *(*) f rj(t) \ 0 fora

14) by parts. Since 77(01) == 0 = 77(02), we obtain 2 / 0(t,r7,7))dt= / (