[Article] Hamiltonians and Lagrangians of non-autonomous one-dimensional mechanical system

[Article] Hamiltonians and Lagrangians of non-autonomous one-dimensional mechanical system

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INVESTIGACI ´ ON REVISTA MEXICANA DE F ´ ISICA 52 (5) 429–432 OCTUBRE 2006 Hamiltonians and Lagrangians of non-autonomous one-dimensional mechanical systems G.F. Torres del Castillo Departamento de F´ ısica Matem´atica, Instituto de Ciencias, Universidad Aut´onoma de Puebla, 72570 Puebla, Pue., M´exico. I. Rubalcava Garc´ ıa Facultad de Ciencias F´ ısico Matem´aticas, Universidad Aut´onoma de Puebla, Apartado Postal 1152, 72001 Puebla, Pue., M´exico. Recibido el 25 de mayo de 2006; aceptado el 12 de septiembre de 2006 It is shown that a given non-autonomous system of two first-order ordinary differential equations can be expressed in Hamiltonian form. The derivation presented here allows us to obtain previously known results such as the infinite number of Hamiltonians in the autonomous case and the Helmholtz condition for the existence of a Lagrangian. Keywords: Non-autonomous systems; Hamilton equations; Lagrangians. Se muestra que un sistema dado, no aut´onomo, de ecuaciones diferenciales ordinarias de primer orden puede expresarse en forma hamil- toniana. La deducci´on presentada aqu´ ı nos permite obtener resultados previamente conocidos tales como el n´umero infinito de hamiltonianas en el caso aut´onomo y la condici´on de Helmholtz para la existencia de una lagrangiana. Descriptores: Sistemas no aut´onomos; ecuaciones de Hamilton; lagrangianas. PACS: 45.05.+x; 45.20.-d 1. Introduction As is well known, it is very convenient to express a given sys- tem of ordinary differential equations (not necessarily related to classical mechanics) as the Euler–Lagrange equations as- sociated with some Lagrangian, L , or as the Hamilton equa- tions associated with some Hamiltonian, H (see, e.g. , Ref. 1). One of the advantages of such identifications is the possibility of finding constants of motion, which are related to symme- tries of L or H . Also, the Hamiltonian of a classical system is usually regarded as an essential element to find a quantum version of the mechanical system. In the simple case of a mechanical system with forces derivable from a potential (that may depend on the veloci- ties), there is a straightforward procedure for finding a La- grangian or a Hamiltonian. However, in the case of non- conservative mechanical systems or of systems not related to mechanics, the problem of finding a Lagrangian or a Hamil- tonian is more involved. A given system of n second-order ordinary differential equations is equivalent to the Euler– Lagrange equations for some Lagrangian if and only if a set of conditions (known as the Helmholtz conditions) are ful- filled (see, e.g. , Refs. 2, 3, and the references cited therein). The aim of this paper is to give a straightforward pro- cedure to find a Hamiltonian for a given system of two first-order ordinary differential equations (which may not be equivalent to a second-order ordinary differential equation) that possibly involves the time in an explicit form. The re- sults derived here contain the Helmholtz condition for
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[Article] Hamiltonians and Lagrangians of non-autonomous one-dimensional mechanical system

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