[Article] The Lagrangians Of A One-Dimensional Mechanical System

# [Article] The Lagrangians Of A One-Dimensional Mechanical System

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Unformatted text preview: INVESTIGACI ´ ON REVISTA MEXICANA DE F ´ ISICA 50 (4) 379–381 AGOSTO 2004 The Lagrangians of a one-dimensional mechanical system G.F. Torres del Castillo Departamento de F´ ısica Matem´atica, Instituto de Ciencias, Universidad Aut´onoma de Puebla, Apartado Postal 1152, 72001 Puebla, Pue., M´exico Recibido el 23 de junio de 2003; aceptado el 19 de febrero de 2004 Starting from the fact that for an arbitrary autonomous mechanical system any constant of motion can be used as Hamiltonian, the expression for the Lagrangians of a one-dimensional mechanical system previously found by other authors is derived. Keywords: Lagrangians; Hamilton equations. Partiendo del hecho de que para un sistema mec´anico aut´onomo arbitrario cualquier constante de movimiento puede usarse como hamiltoni- ana, se deduce la expresi´on para las lagrangianas de un sistema mec´anico unidimensional previamente hallada por otros autores. Descriptores: Lagrangianas; ecuaciones de Hamilton. PACS: 45.05.+x; 45.20.-d 1. Introduction For a given one-dimensional autonomous dynamical system in classical mechanics, there are infinitely many Lagrangians; in fact, from each constant of motion of the system a La- grangian can be obtained. Specifically, if K ( q, ˙ q ) is a con- stant of motion of the system, a Lagrangian is explicitly given by [1-4] L ( q, ˙ q ) = ˙ q ˙ q Z K ( q,y ) y 2 dy (1) and, as can be readily verified, the corresponding Hamilto- nian is H ( q,p ) = K ( q, ˙ q ( q,p )) , assuming that the relation between the canonical momentum and ˙ q can be inverted, i.e. , the constant of motion employed in the construction of the Lagrangian is essentially the Hamiltonian. In a recent paper [5] a somewhat similar result has been established. For an autonomous system with n degrees of freedom, and forces derivable from a potential, any constant of motion can be employed as Hamiltonian of the system, provided that the Poisson bracket is suitably defined, and if n > 1 , there are infinitely many suitable Poisson brackets for each Hamiltonian. The aim of this paper is to show that in the case where n = 1 these two approaches are equivalent to each other; following the procedure given in Ref. 5 we derive Eq. (1)....
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[Article] The Lagrangians Of A One-Dimensional Mechanical System

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