Parameter-invariant problems of higher order in the calculus of variations have certain characteristic properties which give rise to considerable difficulties in any attempt to apply the well-known methods to develop a theory of conjugate points. In this paper a direct method is thus developedab initio which generalizes the approach of Bliss to the parameter-invariant first order problem. The Jacobi equations for the second order problem (to which the attention is initially restricted) are defined as the Euler-Lagrange equations of the accessory problem (which is non-parameter-invariant). Some properties of the Jacobi equations are investigated; in particular, so-called normal solutions of these differential equations are defined, and it is shown that all the usual existence theorems do apply to them. This fact is then used in the proof of the necessity condition of Jacobi, which is stated as usual in terms of conjugate points. While originally two equivalent definitions of the conjugate points are given in terms of solutions of the Jacobi equations these points may also be determined analytically as zeros of a certain determinant. This result is used to show that the theory is nevertheless parameter-invariant despite the’ non-parameter-invariance of the accessory problem. The final section is devoted to a description of the basic concepts of the general theory for the parameter-invariant problem of arbitrary order.
G. A. Bliss,Jacobi’s condition for problems of the calculus of variations in parametric form, Trans. Amer. Math. Soc.17 (1916), pp. 195–206.
G. A. Bliss,Lectures on the calculus of variations, University of Chicago Press, Chicago (1946).
A.R. Forsyth,Calculus of variations, Cambridge (1927). Dover reprint, New York (1960).
P. Funk,Variationsrechnung und ihre Anwendung in Physik und Technik, Springer-Verlag. Berlin-Göttingen-Heidelberg (1962).
H.S.P. Grässer,Die Transversalitäts-und Brennpunktbedingungen bei Variationsproblemen zweiter Art mit veränderlichen Endpunkten, Tydskr. Natuurwet.5 (1965), pp. 196–214.
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).
H.S.P. Grässer,An imbedding theorem for parameter-invariant higher order problems in the calculns of variations, Ann. Mat. Pura Appl. 4 77 (1967), pp. 377–394.
C.G.J. Jacobi,Zur Theorie der Variationsrechnung und der Differentialgleichungen, Journal für Mathematik17 (1837), pp. 68–82.
E. Kamke,Differentialgleichungen I. Gewöhnliche Differentialgleichungen, 5.Auflage, Akademische Verlagsgesellschaft Geest & Portig, Leipzig (1964).
L.H. McFarlan,The theory of problems in the calculus of variations in several dependent variables and their derivatives of various orders, Tôhoku Math. J.33 (1931), pp. 204–218.
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.
The results presented in sections 1 to 7 of this article are contained in a doctoral thesis which was submitted to the University of South Africa. The writer wishes to thank his supervisor, ProfessorH. Rund, for his valuable criticism and advice concerning the thesis as well as the present article.
About this article
Cite this article
Grässer, H.S.P. On the theory of conjugate points for parameter-invariant higher order problems in the calculus of variations. Annali di Matematica 79, 71–92 (1968). https://doi.org/10.1007/BF02415179
- Differential Equation
- General Theory
- Basic Concept
- Final Section
- Characteristic Property