The coarse-grained dynamical equations may either have or not have reversible terms giving rise to inertial coupling between the polarization ψ (that is, the generalized coordinate) and the spin s (that is, the generalized momentum). In the first case ( g ≠ 0) we have an inertial theory, with a Poisson structure expressing the fact that s is the generator of the rotational symmetry, thus leading to conservation of the total spin. In the second case ( g = 0) we recover the non-inertial theory of ref. 18 , where polarization is decoupled from the spin and the symmetry does not entail any Poisson structure (the equation for s becomes irrelevant). In this case z = 1.73. On the other hand, in the inertial theory, the irreversible kinetic coefficient of the spin may be either conservative or non-conservative. In the conservative case there is no spin dissipation ( η = 0), which produces the inertial-conservative fixed point with z = 1.35. In the non-conservative case, the kinetic coefficient contains a dissipative term ( η ≠ 0), although the impact of dissipation depends on how strong that is compared to the system size. In the underdamped regime, η L z ≪ 1, collective fluctuations are still ruled by the inertial-conservative fixed point, so z = 1.35. This is the regime of natural swarms. Conversely, in the overdamped regime, η L z ≫ 1, the Poisson structure is washed out, the spin drops out of the calculation, and collective fluctuations are ruled by the fully non-conservative fixed point, hence giving z = 1.73.
