The Newton Form of the Geodesic Flow on S R 2 and E A,B 2 in Maupertuis Gauge

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 2R with the radius R and on the biaxial ellipsoid E 2A,B 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. The twodimensional Riemann manifold is represented by conformal coordinates (isometric coordinates, isothermal coordinates), 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. KeywordsNewton MechanicGeodesic FlowLocal ChartUniversal Transverse MercatorMinimal GeodesicThese 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.