×

Composite variational principles, added variables, and constants of motion. (English) Zbl 0608.34053

It is shown that any second order differential system admits a variational formulation via the introduction of suitable additional variables. The new variables are related to the existence of invariant 1- forms and to solutions for the adjoint of the equations of variation of the given system. The connections among invariant forms, constants of motion, and infinitesimal invariance transformations are then discussed in some detail.

MSC:

34C99 Qualitative theory for ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antonini, P., Marmo, G., and Rubano, C. (1985).Nuovo Cimenta B,86, 17.
[2] Atherton, R. W., and Homsy, G. M. (1975).Studies in Applied Mathematics,54, 31.
[3] Bahar, L. Y., and Kwatny, H. G. (1984).Mechanics Research Communications,11, 253. · Zbl 0565.70020
[4] Carinena, J. F., and Ibort, L. A. (1985).Physics Letters,107A, 356.
[5] Caviglia, G. (1983a).Journal of Mathematical Physics,24, 2065. · Zbl 0532.70014
[6] Caviglia, G. (1983b).International journal of Theoretical Physics,22, 1051. · Zbl 0563.70015
[7] Caviglia, G. (1984).International Journal of Theoretical Physics,23, 461.
[8] Caviglia, G. (1985a).Inverse Problems,1, L13. · Zbl 0593.70002
[9] Caviglia, G. (1985b).International Journal of Theoretical Physics,24, 377. · Zbl 0571.70017
[10] Crampin, M. (1980).Physics Letters,79A, 138.
[11] Crampin, M. (1981).Journal of Physics A: Mathematical Nuclear and General,14, 2567. · Zbl 0475.70022
[12] Crampin, M. (1983).Physics Letters,95A, 209.
[13] De Filippo, S., Vilasi, G., Marmo, G., and Salerno, M. (1984).Nuovo Cimenta B,83, 97.
[14] Gonzalez-Gascon, F., and Rodriguez-Camino, E. (1980).Letter al Nuovo Cimento,29, 113.
[15] Greenberger, D. M. (1979).Journal of Mathematical Physics,20, 762.
[16] Hojman, S., and Urrutia, R. F. (1981).Journal of Mathematical Physics,22, 1896. · Zbl 0475.70023
[17] Leipholz, H. H. E. (1980). InTheoretical and Applied Mechanics. North-Holland, Amsterdam, p. 1.
[18] Morse, P. M., and Feshbach, H. (1953).Methods of Theoretical Physics, Vol. 1. McGraw-Hill, New York, p. 298. · Zbl 0051.40603
[19] Prince, G. E., and Crampin, M. (1984).General Relativity Gravitation,16, 921. · Zbl 0556.53044
[20] Santilli, R. M. (1978).Foundations of Theoretical Mechanics I. The Inverse Problem in Newtonian Mechanics. Springer, New York. · Zbl 0401.70015
[21] Sarlet, W., and Cantrijn, F. (1981).Siam Review,23, 467. · Zbl 0474.70014
[22] Schattner, R., and Trumper, M. (1981).Journal of Physics A: Mathematics General and Nuclear,14, 2345.
[23] Thangaray, D., and Venkatarangan, S. N. (1983).IMA Journal of Applied Mathematics,30, 21. · Zbl 0515.49008
[24] Tikochinsky, Y. (1978).Journal of Mathematical Physics,19, 888.
[25] Trumper, M. (1980).Comptes Rendus Hebdomadaires des Seances de l’ Academie des Sciences Paris,291, 615.
[26] Trumper, M. (1983).Annals of Physics (New York),149, 203. · Zbl 0522.70021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.