Nambu-Poisson mechanics and its applications

Abstract

Nambu mechanics is a generalization of Hamiltonian mechanics. In the standard Hamiltonian mechanics the equations of motion are formulated in terms of the Poisson bracket and one Hamiltonian. However, some physical systems like the motion of the electrically charged particle in the field of magnetic monopole does not admit a satisfactory description within the Hamiltonian formalism. The Nambu mechanics is an alternative to the Hamiltonian formalism. In this picture the evolution is described in terms of the Nambu bracket defined for three arguments (f,g,h) and two Hamiltonians. The quantization of the corresponding mechanical system is an open and challenging problem. Our aim is to construct new non-trivial examples of physical systems which can be treated within the framework of Nambu mechanics. Furthermore, this formalism can be a suitable introduction to the mathematical structure present in non-associative quantum mechanics. The latter being object of great interest from the scientific community due to the non-associative structures present in string theory.