Poisson structures for the Aristotelian model of three-body motion

We present explicitly Poisson structures of a dynamical system with three degrees of freedom introduced and studied by Calogero et al (2005 J. Phys. A: Math. Gen. 38 8873-96). We first show the construction of a formal Hamiltonian structure for a time-dependent Hamiltonian function. We then cast the dynamical equations into the form of a gradient flow by means of a potential function. By reducing the number of equations, we obtain the second time-independent Hamiltonian function which includes all parameters of the system. This extends the result of Calogero et al (2009 J. Phys. A: Math. Theor. 42 015205) for semi-symmetrical motion. We present bi-Hamiltonian structures for two special cases of the cited references. It turns out that the case of three bodies two of which are not interacting with each other but are coupled through the interaction of a third one requires a separate treatment. We conclude with a discussion on generic form of the second time-independent Hamiltonian function. © 2011 IOP Publishing Ltd.