Advertisement

An imbedding theorem for parameter-invariant higher order problems in the calculus of variations

  • 16 Accesses

  • 1 Citations

Summary

Certain characteristic properties of parameter-invariant problems of the second order in the calculus of variations obstruct a direct approach to the basic imbedding theorem. Such a theorem may nevertheless be derived by the application of a generalization of a method due to Bliss for the first order problem. In the case of second order problems an extremal is uniquely determined by line elements of the type\((x^i , \dot x^i , \ddot x^i , \dddot x^i )\), it being assumed that the matrix of the second derivatives of the Lagrangian with respect to\(\ddot x^i \) has rank n − 1. Such an extremal may always be imbedded in a (4n − 4) -parameter family of extremals. A certain determinant, which bears a close relationship to the Mayer determinant of the non-parameter-invariant problem, is defined in terms of such a family of extremals, and it is found that this determinant does not vanish along such extremals. Similar results may be obtained for parameter-invariant problems of arbitrary order.

References

  1. [1]

    G. A. Bliss,Lectures on the calculus of variations, University of Chicago Press, Chicago (1946).

  2. [2]

    H. V. Craig,On the solution of the Euler equations for their highest derivatives, Bull. Amer. Math. Soc.36 (1930), pp. 558–562.

  3. [3]

    H.S.P. Grässer,Hamilton-Jacobi theory for parameter-invariant problems of the second order in the calculus of variations, Ph. D. thesis, University of South Africa, Pretoria (1966).

  4. [4]

    E. Kamke,Differentialgleichungen 1,Gewöhnliche Differentialgleichungen, 5. Auflage, Akademische Verlagsgesellschaft Geest & Portig, Leipzig (1964).

  5. [5]

    L. H. McFarlan,Problems of the calculus of variations in several dependent variables and their derivatives of various orders, Tôhoku Math. J.33 (1931), pp. 204–218.

  6. [6]

    H. Rund,The theory of problems in the calculus of variations whose Lagrangian function involves second order derivatives: A new approach, Ann. Mat. Pura Appl. (4) 55 (1961), pp. 77–104.

  7. [7]

    —— ——,Canonical formalism for parameter-invariant integrals in the calculus of variations whose Lagrange functions involve second order derivatives, Ann. Mat. Pura Appl. (4)63 (1964), pp. 99–107.

  8. [8]

    —— ——,The Hamilton-Jacobi theory in the calculus of variations, Van Nostrand, London-New York (1966).

Download references

Author information

Additional information

Most of the results of this paper (§§1 – 3) are contained in a doctoral thesis ([3]) which was presented to the University of South Africa. The writer wishes to express his gratitude to his supervisor, ProfessorH. Rund, for his interst, encouragement and advice concerning the thesis and the present article.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Grässer Pretoria, H.S.P. An imbedding theorem for parameter-invariant higher order problems in the calculus of variations. Annali di Matematica 77, 377–394 (1967). https://doi.org/10.1007/BF02416950

Download citation

Keywords

  • Characteristic Property
  • Line Element
  • Direct Approach
  • Parameter Family
  • Arbitrary Order