A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem

<p style='text-indent:20px;'>In this paper, we prove a symmetric property for the indices for symplectic paths in the enhanced common index jump theorem (cf. Theorem 3.5 in [<xref ref-type="bibr" rid="b6">6</xref>]). As an application of this property, we prove that on every compact Finsler manifold <inline-formula><tex-math id="M1">\begin{document}$ (M, \, F) $\end{document}</tex-math></inline-formula> with reversibility <inline-formula><tex-math id="M2">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and flag curvature <inline-formula><tex-math id="M3">\begin{document}$ K $\end{document}</tex-math></inline-formula> satisfying <inline-formula><tex-math id="M4">\begin{document}$ \left(\frac{\lambda}{\lambda+1}\right)^2&lt;K\le 1 $\end{document}</tex-math></inline-formula>, there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form <inline-formula><tex-math id="M5">\begin{document}$ e^{\sqrt {-1}\theta} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \frac{\theta}{\pi}\notin{\bf Q} $\end{document}</tex-math></inline-formula> provided the number of closed geodesics on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> is finite.</p>

The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid

In this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.

Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

Fiberwise convexity of Hill’s lunar problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.