Algebraic Closed Geodesics on a Triaxial Ellipsoid

Details Algebraic Closed Geodesics on a Triaxial Ellipsoid Algebraic Closed Geodesics on a Triaxial Ellipsoid

2005-06-29

arxiv.org/abs/nlin/0506063v2

Regular and Chaotic Dynamics, 10, No.4 (2005), 463-485

Nonlinear Sciences

Exactly Solvable and Integrable Systems

15 figures

Scientific paper

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in ${\mathbb R}^3$. Such geodesics are either connected components of spatial elliptic curves or rational curves. Our approach is based on elements of the Weierstrass--Poncar\'e reduction theory for hyperelliptic tangential covers of elliptic curves and the addition law for elliptic functions. For the case of 3-fold and 4-fold coverings, explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.

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