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Active matter is a priori composed of entities which produce a work and are therefore fundamentally out of equilibrium. The energy is not conserved, the momentum of the particles either. Finally, the very fact that the particles have a reference velocity breaks the Galilean invariance. Are all these effects necessary simultaneously to observe the collective motions observed in active matter? In order to explore this question, we proposed a model for a fluid of spins at equilibrium, which conserves energy and momentum, but whose Lagrangian contains terms that couple spins and velocity. As a result, the momentum is not the mass times the velocity and the Galilean invariance is broken. Consequently, it is possible in principle to develop collective motion. This is indeed what is found according to a mean field calculation. On the other hand, in finite dimension, the mean field solutions are unstable and leave space for solutions of vortex and solitons. Bore, S. L., Schindler, M., Lam, K.-D. N. T., Bertin, E. M., & Dauchot, O. (2016). Coupling spin to velocity: collective motion of Hamiltonian polar particles. Journal of Statistical Mechanics: Theory and Experiment, 2016(3), 033305. |