The Gaussian curvature of associated manifold of dynamical systems

Details The Gaussian curvature of associated manifold of dynamical systems The Gaussian curvature of associated manifold of dynamical systems

Aug 1984

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1984ap%26ss.103...99e&link_type=abstract

Astrophysics and Space Science (ISSN 0004-640X), vol. 103, no. 1, Aug. 1984, p. 99-113.

Mathematics

1

Celestial Mechanics, Equations Of Motion, Ergodic Process, Gauss Equation, Manifolds (Mathematics), Orbital Mechanics, Geodesic Lines, Newton Theory, Periodic Functions, Poincare Problem, Potential Fields, Toroids, Trajectory Analysis

Scientific paper

Manifolds with Gaussian curvature are considered, taking into account the importance of manifolds with positive curvature in astronomy and astrophysics. Orbits in a Newtonian potential, orbits in magnetic fields, at least in first-order approximation, and the behavior of charged particles in magnetic confinement devices can be translated into terms of behavior of geodesics on a surface with a metric defined by means of the potential of the system. The present investigation is concerned with the study of dynamical systems governed by a potential v. The manifold on which the geodesics are defined is referred to as the 'associated manifold'. The study is mainly conducted in terms of Jacobi's geodesic deviation. Three areas of applications are distinguished, giving attention to dynamical systems with mixed type of Gaussian curvature, toroidal manifolds where the Gaussian curvature is a periodic function of the poloidal angle, and dynamical systems with a nonnegative Gaussian curvature of their associated manifold.

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