Brachistochrone in almost flat space

Journal Article · Sat Oct 01 00:00:00 EDT 1988 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.528158· OSTI ID:6904590

Kamath, S G

This paper is an extension, within the framework of general relativity, of the relativistic brachistochrone discussed recently by Goldstein and Bender (J. Math. Phys. 27, 507 (1986)). Assuming that the gravitational field due to a spherically symmetric source with mass M at equilibrium is weak, it is found that the brachistochrone, for a falling particle of mass m, described by THETA(r), with THETA an angle and r a distance measured from the center of symmetry, is in general a hyperelliptic integral. The latter integral can in one case be calculated exactly in terms of the normal elliptic integrals of the first and third kinds and the elementary transcendental functions. It is shown via a numerical computation using the sun's gravitational field as a reference that one can recast this exact version into a simple form, viz., (r)/sup 1/2/THETA = a, where a is a constant.

• Research Organization: Department of Mathematics, Indian Institute of Technology, Madras 600 036, India
• OSTI ID: 6904590
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 29:10; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
DISTRIBUTION
EQUATIONS
EQUATIONS OF MOTION
FIELD THEORIES
FUNCTIONS
GENERAL RELATIVITY THEORY
GRAVITATION
GRAVITATIONAL FIELDS
INTEGRALS
MASS DISTRIBUTION
METRICS
PARTIAL DIFFERENTIAL EQUATIONS
POLYNOMIALS
SPATIAL DISTRIBUTION
SYMMETRY