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New solutions to the central force problem: a class of generalized conic trajectories

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Abstract

New solutions to the central force problem are detailed. These solutions correspond to a class of curves which is obtained by the generalization of a geometric property of conics. This class contains a wide variety of well-known curves and in particular admits the logarithmic spiral as a limiting case. The determination of the law of central force is made by using Siacci’s theorem. This law allows us to link in a single relation the linear, inverse square and inverse cubic forces. New trajectories and corresponding analytic solutions are provided. The results are relevant to the fields of celestial mechanics and orbital dynamics.

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Data used in the paper are given in the reference part: MATHCURVE.COM. Cardioide (https://mathcurve.com/courbes2d/cardioid/cardioid.shtml). Construction de la tangente en un point—GeoGebra (https://www.geogebra.org/m/uQUTjSvP#material/mvvQjNVj). Construction à la règle et au compas de la tangente à une ellipse, sans utiliser les foyers (pagesperso-orange.fr) http://alain.pichereau.pages.perso-orange.fr.

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Acknowledgements

The author wishes to thank the anonymous reviewer for constructive criticism and stimulating observations. His (her) contribution was important in improving the article.

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  1. 5 Rue Romarin, 69001, Lyon, France

    Eric Guiot

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  1. Eric Guiot
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Correspondence to Eric Guiot.

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A part of the results are prepublished under another title on the web site HAL Dynamics of Generalized Conic Trajectories - Archive ouverte HAL (archives-ouvertes.fr).

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Guiot, E. New solutions to the central force problem: a class of generalized conic trajectories. Meccanica 58, 1511–1522 (2023). https://doi.org/10.1007/s11012-023-01682-1

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