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We propose a conservative two-dimensional particle model in which particlescarry a continuous and classical spin. The model includes standardferromagnetic interactions between spins of two different particles, and anonstandard coupling between spin and velocity of the same particle inspired bythe coupling observed in self-propelled hard discs. Because of this couplingGalilean invariance is broken and the conserved linear momentum associated totranslation invariance is not proportional to the velocity of the center ofmass. Also, the dynamics is not invariant under a global rotation of the spinsalone. This, in principle, leaves room for collective motion and thus raisesthe question whether collective motion can arise in Hamiltonian systems. Westudy the statistical mechanics of such a system, and show that, in the fullyconnected (or mean-field) case, a transition to collective motion does exist inspite of momentum conservation. Interestingly, the velocity of the center ofmass, which in the absence of Galilean invariance, is a relevant variable, alsofeeds back on the magnetization properties, as it acts as an external magneticfield that smoothens the transition. Molecular dynamics simulations of finitesize systems indeed reveal a rich phase diagram, with a transition from adisordered to a homogeneous polar phase, but also more complex inhomogeneousphases with local order interrupted by topological defects.

Coupling spin to velocity: collective motion of Hamiltonian polar particles