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The dependence of focal points upon curvature for problems of the calculus of variations in space

1912
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Transactions of the American Mathematical Society
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In the Calculus of Variations, if an arc Coi which joins a space curve L and a fixed point 1 minimizes the integral (1) J = J f(x, y,y', z, z')dx with respect to other curves joining L with 1, there will in general be a focal point 2 lying beyond 1 on the curve C of which C0i is a part, at which the minimizing property ceases. For the space problem it is well known that the minimizing arc Coi must an extremal and must be cut by L transversally at their point of intersection 0 .f If these

doi:10.1090/s0002-9947-1912-1500914-4
fatcat:qxggdrxl35gg5bnvfh7jodnwh4