[Article] Sympletic Structures And Hamiltonians Of A Mechanical System

[Article] Sympletic Structures And Hamiltonians Of A Mechanical System

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Unformatted text preview: INVESTIGACI ON REVISTA MEXICANA DE F ISICA 49 (5) 445449 OCTUBRE 2003 Symplectic structures and Hamiltonians of a mechanical system G.F. Torres del Castillo Departamento de F sica Matematica, Instituto de Ciencias, Universidad Autonoma de Puebla, 72570 Puebla, Pue., Mexico G. Mendoza Torres Facultad de Ciencias F sico Matematicas, Universidad Autonoma de Puebla, Apartado Postal 1152, 72001 Puebla, Pue., Mexico Recibido el 6 de mayo de 2003; aceptado el 12 de junio de 2003 It is shown that in the case of a mechanical system with a finite number of degrees of freedom in classical mechanics, any constant of motion can be used as Hamiltonian by defining appropriately the symplectic structure of the phase space (or, equivalently, the Poisson bracket) and that for a given constant of motion, there are infinitely many symplectic structures that reproduce the equations of motion of the system. Keywords: Symplectic structure; Hamilton equations. Se muestra que en el caso de un sistema mecanico con un numero finito de grados de libertad en la mecanica clasica, cualquier constante de movimiento puede usarse como hamiltoniana definiendo apropiadamente la estructura simplectica del espacio fase (o, equivalentemente, el parentesis de Poisson) y que para una constante de movimiento dada, existe una infinidad de estructuras simplecticas que reproducen las ecuaciones de movimiento del sistema. Descriptores: Estructura simplectica; ecuaciones de Hamilton. PACS: 45.05.+x; 45.20.-d 1. Introduction The equations of motion of a mechanical system with a fi- nite number of degrees of freedom in classical mechanics are usually the EulerLagrange equations for the Lagrangian L = T- U , where T denotes the kinetic energy and U is the potential energy (assuming that the forces are derivable from a potential). The equations of motion can also be expressed in the form of Hamiltons equations, which are equivalent to df/dt = { f,H } , for any function f that does not depend ex- plicitly on the time, defined on the phase space, where H is the Hamiltonian and { , } is the Poisson bracket. The Hamil- tonian is usually obtained from the Lagrangian by means of the Legendre transformation and, frequently, but not always, H corresponds to the total energy (see, e.g. , Refs. 1, 2). As shown below, for a given mechanical system with a fi- nite number of degrees of freedom there are infinitely many Hamiltonians, which need not be derived from a Lagrangian, and for each choice of the Hamiltonian, there are infinitely many Poisson brackets that allow us to express the equations of motion in Hamiltonian form (see also Refs. 35). In Sec. 2 the basic theory is reviewed. In Sec. 3 some concrete examples are given, considering two simple systems with two degrees of freedom, and in Sec. 4 the general re- sults are established. Throughout this paper the summation convention is employed....
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[Article] Sympletic Structures And Hamiltonians Of A Mechanical System

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