Time transformation and reversibility of Nambu--Poisson systems

Abstract

A time transformation technique for Nambu--Poisson systems is developed, and its structural properties are examined. The approach is based on extension of the phase space $\mathscr{P}$ into $\overline{\mathscr{P}}=\mathscr{P}\times\mathbb{R}$, where the additional variable controls the time-stretching rate. It is shown that time transformation of a system on $\mathscr{P}$ can be realised as an extended system on $\overline{\mathscr{P}}$, with an extended Nambu-Poisson structure. In addition, reversible systems are studied in conjunction with the Nambu-Poisson structure. The application in mind is adaptive numerical integration by splitting of Nambu-Poisson Hamiltonians. As an example, a novel integration method for the rigid body problem is presented and analysed.