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Mathematische Annalen

, Volume 58, Issue 1–2, pp 70–80 | Cite as

Jacobi's criterion when both end-points are variable

  • G. A. Bliss
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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.Google Scholar
  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.Google Scholar
  3. *).
    Problems satisfying the condition c) have been named by Hilbert,regular problems.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  8. *).
    The interior of the dotted lines in the figure. See Bolza, Transactions of the American Mathematical Society, Vol 2, p. 424.Google Scholar
  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.Google Scholar
  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.Google Scholar

Copyright information

© Springer-Verlag 1903

Authors and Affiliations

  • G. A. Bliss
    • 1
  1. 1.Göttingen

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