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Relativistic brachistochrone

1986; American Institute of Physics; Volume: 27; Issue: 2 Linguagem: Inglês

10.1063/1.527199

1527-2427

Harris F. Goldstein, Carl M. Bender,

Sports Dynamics and Biomechanics

The trajectory joining two points a1 and a2, which minimizes the transit time for a particle, initially at rest, to fall in a uniform gravitational field from a1 to a2, is called the brachistochrone. Johann Bernoulli was the first to find an analytical form for the brachistochrone; in 1696, he discovered that the trajectory is a cycloid. In this paper the relativistic generalization of this classic problem is presented. Four separate curves are actually identified: a particle falling in both a uniform electric and uniform gravitational field is considered. The curves that minimize the times of flight measured by an observer in a laboratory in which a1 and a2 are fixed and also the curves that minimize the proper times of flight are found.