By Werner Hildbert Greub, Stephen Halperin, James Van Stone

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The result at the points where u jumps then follows by taking appropriate limits, and likewise at to — a, b. We then consider 0 < e < min(£o — a, b — to) and define r? 3 The second variation. Jacobi fields for given ( e l f 21 Then { 0 £ -£ for a < t < t0 or t0 + e < t < b for t0 - e < t < t0 for t0 < t < t0 + e. 1 to obtain 0 < 62I(u, rj) = / ° Fpipj (t, u(t), u{t))CZJdt + 0(e 2 ) for c -+ 0, Jto-e since all other terms contain a factor e, and we integrate over an interval of length 2e. Zj = lim - / Fpipj{t,u{t),ii(t))Ctjdt > 0.

N . In particular, inserting t = 0, we get r)k(0)vjvk = 0 for all v G R n , i = 1 , . . , n. We use t; = e*, where (ei)l=1 n is an orthonormal basis of R n . Then r{i(0) = 0 for a l i i and/. We next insert v = ^(ej 4- e m ), £ ^ m. The symmetry Tljk — T^. (which directly follows from the definition of H fc and the symmetry gjk = #fcj) then yields rj m (0) = 0 for a l l i , / , m . The vanishing of gij^ for all i,j, k then is an easy exercise in linear algebra. d. 2. 4 are called Riemannian normal coordinates.

We then consider 0 < e < min(£o — a, b — to) and define r? 3 The second variation. Jacobi fields for given ( e l f 21 Then { 0 £ -£ for a < t < t0 or t0 + e < t < b for t0 - e < t < t0 for t0 < t < t0 + e. 1 to obtain 0 < 62I(u, rj) = / ° Fpipj (t, u(t), u{t))CZJdt + 0(e 2 ) for c -+ 0, Jto-e since all other terms contain a factor e, and we integrate over an interval of length 2e. Zj = lim - / Fpipj{t,u{t),ii(t))Ctjdt > 0. d The Jacobi equations and the notion of Jacobi fields are meaningful for arbitrary solutions of the Euler-Lagrange equations, not only for minimizing ones.