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Generalizations of the Brachistochrone Problem

J. Gemmer·R. Umble·M. Nolan
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Abstract

Consider a frictionless surface S in a gravitational field that need not be uniform. Given two points A and B on S, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This paper considers this problem on simple surfaces such as surfaces of revolution and solves the problem two ways: First, we use conservation of mechanical energy and the Euler-Lagange equation; second, we use geometrical optics and the eikonal equation. We conclude with a discussion of the relativistic effects at relativistic velocities.

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J. Gemmer, R. Umble, M. Nolan. 2009-01-27. Generalizations of the Brachistochrone Problem. https://arxiv.org/abs/math-ph/0612052

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