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Jacobi's criterion when both end-points are variable

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References

  1. Zeitschrift für Mathematik und Physik, Bd. 23 (1878), p. 369. In this article Erdmann also discussed the second variation for the general problem with variable endpoints.

  2. In an exceptional case,d ande may coincide. The analogous exception in the general problem with fixed endpoints, is when the endpoints are conjugates. See Osgood, Transactions of the American Mathematical Society, Vol 2, p. 166.

  3. Problems satisfying the condition c) have been named by Hilbert,regular problems.

  4. See for example, Kneser, Variationsrechnung, pp. 22, 30: Osgood, Sufficient Conditions in the Calculus of Variations, Annals of Mathematics, 2d Ser., Vol. 2, p. 105.

  5. See Kneser, l. c. Variationsrechnung, Sufficient Conditions in the Calculus of Variations, Annals of Mathematics, 2d Ser., Vol. 2 pp. 89, 97; Bliss, Transactions of the American Mathematical Society, Vol. 3, p. 132.

  6. Bliss, l. c. Transactions of the American Mathematical Society, Vol. 3, p. 132. The notation of the present article is slightly different,P, Q, R being used in place ofP 1,P 2,Q.

  7. For a proof that such a set can be determined, see Kneser l. c. Transactions of the American Mathematical Society, Vol. 3, p. 109.

  8. The interior of the dotted lines in the figure. See Bolza, Transactions of the American Mathematical Society, Vol 2, p. 424.

  9. This is a consequence of the field, which is a field forC γ as well as forC, and of the fact thatF y'y'>0 for all values ofy'. See Osgood, Annals, l. c., Variationsrechnung, Sufficient Conditions in the Calculus of Variations, Annals of Mathematics, 2d Ser., Vol. 2, p 119.

  10. Found by substitutingU,V in (2) and eliminating the terms inU,V. The constantc must be different from zero, otherwiseU andV could not be linearly independent. See Jordan, Cours d'Analyse, III, p. 152.

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  1. Göttingen

    G. A. Bliss

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Bliss, G.A. Jacobi's criterion when both end-points are variable. Math. Ann. 58, 70–80 (1903). https://doi.org/10.1007/BF01447780

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