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A REALIZATION OF NAMBU MECHANICS INTERACTING A PARTICLE WITH AN SU 2 MONOPOLE Minoru Hirayama Stanford Linear Accelerator Center Stanford University Stanford California 94305 and Toyama University Toyama 930 Japant ABSTRACT We study the system of a particle degrees of freedom interacting monopole bearing the isospin with an SU 2 t Hooft Polyakov We show that its equation of motion can be cast into the form of Nambu s generalized mechanics Submitted to Phys Rev Comments and Addenda Work supported in part by the Energy Research and Development Administration tPermanent address 2Some time ago Nambu suggested some possible generalizations Hamilunian mechanics As the simplest extension of classical he proposed the replace ment of the conventional canonical doublet in by a set of three variables P Q Rn The usual Poisson bracket was generalized to the Nambu bracket A B C containing three quantities y B C cn Ay y d n n n 1 The time evolution of a dynamical quantity f P Q R was assumed to be determined by 2 f F G where F P Q R and H P Q R are alternatives conventional function in the scheme The appearance of the third variable R makes it difficult which obey Nambu s equations of motion tion for a rigid rotator can be written 2 In these examples to conceive systems It was pointed out that the Euler equa in the form of 2 shown that some systems with constraints mechanics of the Hamiltonian 1 Several authors have can be described by Nambu s the variable R was constructed ventional position and momentum variables from the con In this note we put forth another example of Nambu s mechanics where the variable R cannot be expressed solely as a function of position and momentum variables We consider the classical spin Ti i 1 2 3 interacting Hasenfratz motion of a point particle with mass m and iso with an SU 2 magnetic monopole 3 According to and t Hooft the equations of motion are5 ki m 1 p eAF x Ta 1 3 b fi 1 eAy x Ta 2 eTb p i i 3and qa cabc p 1 eAf x Td eAF TC x 1 and p s are



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