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The Newton Form of the Geodesic Flow on SR2 and EA,B2 in Maupertuis Gauge

  • R. J. You
  • E. Grafarend
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 114)

Abstract

Geodesics, in particular minimal geodesics, are of focal geodetic interest. In the paper we apply the Maupertuis variational principle of least action in the Newton mechanics to transform the geodesic flow on the twodimensional sphere S R 2 with the radius R and on the biaxial ellipsoid E A,B 2 with the semi-major axis A and semi-minor axis B into the Newton form. A geodesic flow on a twodimensional Riemann manifold takes the form of the Newton law if two assumptions are met:
  1. 1.

    The twodimensional Riemann manifold is represented by conformal coordinates (isometric coordinates, isothermal coordinates),

     
  2. 2.
    The arc length s as the curve parameter of a geodesic flow is replaced by the dynamic time t according to the Maupertuis gauge
    $$ds = {\lambda ^2}\left( {{q^1},{q^2}} \right)dt$$
    where A2 is the factor of conformality and q1,q2 are the conformal coordinates which form a local chart of the twodimensional Riemann manifold.
     

Keywords

Newton Mechanic Geodesic Flow Local Chart Universal Transverse Mercator Minimal Geodesic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Goenner, H., E. W. Grafarend and R. J. You: Newton mechanics as geodesic flow on Maupertuis’ manifold: The local isometric embedding into flat spaces, manuscripta geodaetica(199A) in print.Google Scholar
  2. Grafarend, E.: Lecture Notes on map projections (Kartenprojektionen). Department of Geodetic Science, Stuttgart University, 1985.Google Scholar
  3. Grafarend, E. W. and R. J. You: The Newton form of a geodesic in Maupertuis gauge on the sphere and the biaxial ellipsoid: Part I and II. Zeitschrift für Vermessungs wesen(1994) in print.Google Scholar
  4. de Maupertuis, R L. M.: Accord de différentes lois de la Nature. Qui avoient jusqu’ici paru incompatibles, de l’Académie Royal des Sciences de Paris 1744. Reprinted in 1965 Oeuvres Band IV pp. 3–28, Georg Olms Verlagsbuchhandlung, Hildesheim.Google Scholar
  5. Snyder, P. (1987): Map projections — A working manual. U.S. geological survey professional paper 1395, U.S. Government Printing Office, Washington D.C.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • R. J. You
    • 1
  • E. Grafarend
    • 1
  1. 1.Department of Geodetic ScienceStuttgart UniversityStuttgartGermany

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