  Using analytic and numerical methods, we study a $2d$ Hamiltonian model of
interacting particles carrying ferro-magnetically coupled continuous spins
which are also locally coupled to their own velocities. This model has been
characterised at the mean field level in a parent paper. Here, we first obtain
its finite size ground states, as a function of the spin-velocity coupling
intensity and system size, with numerical techniques. These ground states,
namely a collectively moving polar state of aligned spins, and two non moving
states embedded with topological defects, are recovered from the analysis of
the continuum limit theory and simple energetic arguments that allow us to
predict their domains of existence in the space of control parameters. Next,
the finite temperature regime is investigated numerically. In some specific
range of the control parameters, the magnetisation presents a maximum at a
finite temperature. This peculiar behaviour, akin to an order-by-disorder
transition, is explained by the examination of the free energy of the system
and the metastability of the states of minimal energy. The robustness of our
results is checked against the geometry of the boundary conditions and the
dimensionality of space.
