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Solution of the three-dimensional inverse problem

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Abstract

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c 1,g(x, y, z)=c 2, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.

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References

  • Bozis, G.: 1983a,Celest. Mech. 29, 329.

    Google Scholar 

  • Bozis, G.: 1983b,Celest. Mech. 31, 129.

    Google Scholar 

  • Bozis, G.: 1983c,Celest. Mech. 31, 43.

    Google Scholar 

  • Bozis, G.: 1984,Astron. Astrophys. 134, 360.

    Google Scholar 

  • Bozis, G. and Mertens R.: 1985,ZAMM 65, 8, 383.

    Google Scholar 

  • Bozis, G. and Tsarouchas, G.: 1985,Astron. Astrophys. 145, 215.

    Google Scholar 

  • Broucke, R.: 1979,Int. J. Engin. Sci.,17, 1151.

    Google Scholar 

  • Broucke, R. and Lass, H.: 1977,Celest. Mech. 16, 215.

    Google Scholar 

  • Érdi, B.: 1982,Celest. Mech. 28, 209.

    Google Scholar 

  • Favard, J.: 1963,Cours d'Analyse de l'Ecole Polytechnique, Tome 32, Gauthier-Villars, Paris, p. 39.

    Google Scholar 

  • Gascon, F. G. Lopez, A. G., and Broncano P. J. P.: 1984,Celest. Mech. 33, 85.

    Google Scholar 

  • Melis, A. and Piras, B.: 1982, ‘On a Generalization of Szebehely's Problem’, Facoltá di Scienze dell'Università di Cagliari, Vol. LII, facs. 1.

  • Melis, A. and Piras, B.: 1984,Celest. Mech. 32, 87.

    Google Scholar 

  • Mertens, R.: 1981,ZAMM 61, T252-T253.

    Google Scholar 

  • Molnár, S.: 1981,Celest. Mech. 25, 81.

    Google Scholar 

  • Morrison, F.: 1977,Celest. Mech. 16, 39.

    Google Scholar 

  • Puel, F.: 1984,Celest. Mech. 32, 209.

    Google Scholar 

  • Smirnov, V. I.: 1964,A Course of Higher Mathematics, Vol. IV, Pergamon Press, Oxford, p. 359.

    Google Scholar 

  • Szebehely, V.: 1974, in E. Proverbio (ed.),Proceedings of the International Meeting on Earth's Rotation by Satellite Observations, University of Cagliari, Bologna.

    Google Scholar 

  • Szebehely, V. and Broucke, R.: 1981,Celest. Mech. 24, 23.

    Google Scholar 

  • Váradi, F. and Érdi, B.: 1983,Celest. Mech. 30, 395.

    Google Scholar 

  • Whittaker, E. T.: 1944,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover Publ., New York, Ch. IV, p. 96.

    Google Scholar 

  • Xanthopoulos, B. and Bozis, G.: 1983,Astron. Astrohys. 122, 251.

    Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Mechanics, University of Thessaloniki, Greece

    George Bozis & Atef Nakhla

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  1. George Bozis
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  2. Atef Nakhla
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Bozis, G., Nakhla, A. Solution of the three-dimensional inverse problem. Celestial Mechanics 38, 357–375 (1986). https://doi.org/10.1007/BF01238926

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  • DOI: https://doi.org/10.1007/BF01238926

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