Generalizations of the Brachistochrone Problem
نویسندگان
چکیده
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 discusses this problem for simple surfaces such as surfaces of revolution. We first solve this more general problem using the Euler-Lagrange equation and conservation of mechanical energy. We then use geometrical optics to give an alternative method for solving the problem. Finally, we also consider particles that are falling with relativistic velocities.
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