Motion of a falling drop with accretion using canonical methods
G. Hernándeza,* , G. del Vallea,**, I. Camposb and J.L. Jiménezc
ª Área de Física Atómica y Molecular Aplicada, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana–Azcapotzalco, Apartado Postal 21–463, México, D.F., 04000 México, e–mail:* gpe@correo.azc.uam.mx, ** ddg@correo.azc.uam.mx
b Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 21–463, México, D.F., 04000, México, e–mail: iecampos@prodigy.net.mx
c Departamento de Física, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana–Iztapalapa, Apartado Postal 21–463, México D.F., 04000, México, e–mail: jlj@xanum.uam.mx
]]> Recibido el 12 de junio de 2008
Abstract
The motion of a falling drop whose mass grows by accretion is studied with canonical methods. This approach requires the introduction of S–equivalent non natural Lagrangians. That is, we have to consider Lagrangians that give rise to equations of motion that are not exactly the same as the equations of interest, but anyway they share the same solutions. We study three examples of laws of accretion: mass growing linearly with time, mass growing linearly with the surface of the drop, and mass growing proportionally to the product of surface and velocity of the drop. In all cases we recover the results obtained by means of the Newtonian methods, which we expose in table I.
Keywords: Variable mass systems; accretion; S–equivalent Lagrangians; Hamilton–Jacobi formalism.
Resumen
Se estudia, mediante métodos canónicos, la caida de una gota cuya masa crece por acreción. Este enfoque requiere la introducción de Lagrangianos S–equivalentes, no naturales. Esto es, tenemos que considerar Lagrangianos que conducen a ecuaciones de movimiento que no son exáctamente las mismas que las de interés, pero que comparten con ellas las mismas soluciones. Estudiamos tres ejemplos de leyes de acreción: incremento lineal de la masa con el tiempo, incremento de la masa proporcional a la superficie de la gota, e incremento proporcional al producto de la superficie de la gota por la velocidad de la misma. En todos los casos recobramos los resultados obtenidos mediante los métodos Newtonianos, los cuales presentamos en la tabla I.
Descriptores: Sistemas de masa variable; acreción; Lagrangianos S–equivalentes; Formalismo de Hamilton–Jacobi.
]]>PACS: 45.20. Jj; 02.30.Jr
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Acknowledgments
J.L. Jiménez gratefully acknowledges the hospitality of Facultad de Ciencias, where part of this work was prepared during his sabbatical period.
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