The Embedding of the Plumbline Manifold: Orthometric Heights

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

Abstract

The first question the paper attacks is the question whether the plumbline (orthogonal trajectory, field line of the terrestrial gravity field) can be interpreted as a geodesic: (α) If the differential equation x = grad W/∥grad W∥ of a plumbline (W indicates the gravity potential, grad W the gravity vector of Euclidean length ∥grad W∥) is gauged instead of arc length s to A. Marussi time t by means of ds/dt = ∥grad W∥ (A. Marussi,1985.p132) the differential equation of a plumbline reads \({\ddot x^\mu } - \frac{1}{2}{\partial _\mu }{\left\| {grad\,W} \right\|^2} = 0\) as a geodesic in Newton form, (μ = 1,2,3). (β) If the differential equation x = grad W/∥grad W∥ of a plumbline is gauged instead of arc length s to M. Hotine time τ by means of ds/dτ = ∥grad Wf(W) for an arbitrary function f(W) (M. Hotine, 1966,p131–132) the differential equation of a plumbline reads \({\ddot x^\mu } - \frac{1}{2}{\partial _\mu }{\left\| {grad\,W} \right\|^2} \cdot {f^2}(W)) = 0\) as a geodesic in Newton form. (μ = 1,2,3).

Keywords

Manifold Refraction 

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. Hotine, M.: Geodetic applications of conformai transformations, Bulletin Geodesique 80 (1966) 123–140.CrossRefGoogle Scholar
  3. Marussi, A.: Intrinsic geodesy, Springer Verlag, Berlin 1985.CrossRefGoogle Scholar
  4. Moritz, H.: The Hamiltonian structure of refraction and of the gravity field, manuscripta geodaetica (1994) in print.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

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

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