[Article] Making Hidden Symmetries Obvious

[Article] Making Hidden Symmetries Obvious - INVESTIGACION...

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INVESTIGACI ´ ON REVISTA MEXICANA DE F ´ ISICA 54 (2) 127–129 ABRIL 2008 Making hidden symmetries obvious G.F. Torres del Castillo Departamento de F´ ısica Matem´atica, Instituto de Ciencias, Universidad Aut´onoma de Puebla, 72570 Puebla, Pue., M´exico. J.L. Calvario Ac´ocal Facultad de Ciencias F´ ısico Matem´aticas, Universidad Aut´onoma de Puebla, Apartado Postal 1152, 72001 Puebla, Pue., M´exico. Recibido el 12 de septiembre de 2007; aceptado el 21 de febrero de 2008 It is shown that the Hamiltonian of a particle in a uniform gravitational field which possesses a constant of motion not related to transforma- tions in the configuration space, can be expressed in a system of canonical coordinates such that a maximal set of independent constants of motion follows from the existence of ignorable coordinates. Keywords: Hidden symmetries; Hamiltonian formalism. Se muestra que la hamiltoniana de una part´ ıcula en un campo gravitacional uniforme, la cual posee una constante de movimiento no rela- cionada con transformaciones en el espacio de configuraci´on, puede expresarse en un sistema de coordenadas can´onicas tal que un conjunto m´aximo de constantes de movimiento sigue de la existencia de coordenadas ignorables. Descriptores: Simetr´ ıas ocultas; formalismo hamiltoniano. PACS: 45.20.Jj; 03.65.-w; 02.20.Qs 1. Introduction The invariance of the Lagrangian of a mechanical system un- der continuous transformations of the configuration space al- lows one to readily find constants of motion. However, in some cases, there exist constants of motion not related to symmetries in the configuration space. Two well-known ex- amples are the isotropic harmonic oscillator and the Kepler problem; in both cases, in addition to the angular momen- tum, whose conservation follows from the invariance of the standard Lagrangian under rotations about the center of force, there exist constants of motion that are not associated with symmetries of the Lagrangian. It may be remarked that, for a given set of equations of motion, the Lagrangian is not unique (see, e.g. , Ref. 1) and the symmetries of a Lagrangian may not be shared by the alternative Lagrangians. For example, the equations of mo- tion ¨ x = 0 , ¨ y = - g , considered in this paper, can be ob- tained from the Euler–Lagrange equations making use of the usual Lagrangian, given by Eq. (1) below, or the function L 0 = m ˙ x ˙ y - mgx ; in this case one Lagrangian does not depend on x and the other does not depend on y . On the other hand, in the Hamiltonian formalism, ev- ery constant of motion is associated with a symmetry of the Hamiltonian function; in fact, each constant of motion is the infinitesimal generator of a one-parameter group of canoni- cal transformations that leave the Hamiltonian invariant. The
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[Article] Making Hidden Symmetries Obvious - INVESTIGACION...

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