Conserved quantities in the variational equations
C.M. Arizmendi1, J. Delgado2, H.N. Nuñez-Yépez3, A.L. Salas-Brito4*
1 Departamento de Física, Facultad de Ingeniería, Universidad Nacional de Mar del Plata, Mar del Plata, Argentina
2 Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534 Iztapalapa 09340 D.F., México.
3 Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534 Iztapalapa 09340 D.F., México.
4 Laboratorio de Sistemas Dinámicos, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Apartado Postal 21-267, Coyoacán 04000 D.F., México. * Corresponding author
]]>Recibido el 3 de junio de 2002.
Aceptado el 19 de febrero de 2003.
Abstract
Noether's theorem relating continuous symmetries of a Lagrangian system to the existence of conserved quantities is shown to be valid at the level of the variational equations of the system. This result can be helpful in the study of perturbations and of integrability in various areas of current interest. As examples, we derive conserved quatities in linearized general relativity and obtain conserved quantities valid in perturbed classical dynamics.
Keywords: Noether theorem, variational equations, Lagrangian theories.
Resumen
Demostramos que el teorema de Noether, que relaciona simetrías continuas de un sistema lagrangiano con la existencia de cantidades conservadas, es también válido para las ecuaciones variacionales del sistema. Este resultado puede ser de utilidad tanto en la teoría de perturbaciones como en estudios sobre integrabilidad en diversas áreas de interés actual. A guisa de ejemplo encontramos una cantidad conservada en relatividad general mediante el análisis de las simetrías de la gravitación linealizada, y, por otro lado, obtenemos una cantidad conservada muy simple que es válida en la mecánica clásica perturbada.
]]> Palabras clave: Teorema de Noether, ecuaciones variacionales, teorías lagrangianas.PACS: 45.10.Db; 04.20.Fy; 45.20.Jj
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