GENERALIZATIONS OF THE BRACHISTOCHRONE PROBLEM

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 solv...

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Bibliographic Details
Published in:Pi Mu Epsilon journal 2011-04, Vol.13 (4), p.207-218
Main Authors: GEMMER, JOHN A., NOLAN, MICHAEL, UMBLE, RON
Format: Article
Language:English
Subjects:
Brachistochrone curves
Brachistochrone problem
Coordinate systems
Eikonal equation
Electric fields
Force field
Gravitational fields
Mathematical theorems
Refraction
Surfaces of revolution
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Summary: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.
ISSN:0031-952X