Chiral active materials are those that break both time-reversal symmetry and parity microscopically. This is usually a result of the average rotation of the material's complex particles around their center-of-mass (c.m.), which drives the system far from equilibrium. In this paper we show that the nonvanishing spin angular momentum brings about a difference between the c.m. momentum and the total momentum, where the latter accounts for the momentum of all atoms that compose the complex rotating particles. This is in stark contrast to equilibrium fluids where the c.m. stress and the total momentum are essentially equivalent. In fact, we find that generally the c.m. dynamics are insufficient to describe the dynamics of a chiral active material. The total momentum, other than being experimentally accessible in simple rheological experiments, also imposes a constraint—its stress must be allowed to be written in a symmetric way. We find that this requirement imposes a relation between possible central-force interactions and spin-spin interactions, and constraints the amount of odd viscosities in the system to the well-known odd (Hall) viscosity and the odd pressure.

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• Complex fluids
• Nonlinear dynamics in fluids
• Living matter & active matter