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On a tautochrone–related family of paths

 

R. Muñozª, G. Fernández–Anaya^b

 

ª Universidad Autónoma de la Ciudad de México, Centro Histórico, Fray Servando Teresa de Mier 92 y 99, Col. Obrera, Del. Cuauhtémoc, México D.F., 06080, México, e–mail: rodrigo.munoz@uacm.edu.mx

^b Universidad Iberoamericana, Departamento de Física y Matemáticas, Av. Prolongación Paseo de la Reforma 880, Col. Lomas de Santa Fe, Del. Álvaro Obregón México D.F., 01219, México, e–mail: guillermo.fernandez@uia.mx

 

Recibido el 26 de julio de 2010
Aceptado el 30 de agosto de 2010

 

Abstract

An alternative approach to the properties of the tautochrone and brachistochrone curves is used to introduce a family of curves complying with relations where the time of descent is proportional to a fractional power of the height difference. These curves are classified acording with their symmetries. Further properties of these curves are studied.

Keywords: Analytical mechanics; Huygens's isochrone curve; Abel's mechanical problem.

 

Resumen

Utilizamos un tratamiento alternativo de las curvas tautocrona y braquistocrona para introducir una familia de curvas que cumplen con relaciones en las que el tiempo de descenso es directamente proporcional a la altura descendida, elevada a un valor fraccionario. Las mencionadas curvas son clasificadas de acuerdo con sus simetrías. Se estudian otras propiedades de dichas curvas.

Descriptores: Mecánica analítica; curva isocrona de Huygens; problema mecánico de Abel.

 

PACS: 01.55.+b; 02.30Xx; 02.30.Hq; 02.30.Em

 

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