 Cosmological Lagrangian displacement

The cosmological Lagrangian displacement sends an initially labelled matter element from its Lagrangian coordinate $\mathbf{q}$ to $\mathbf{x}(\mathbf{q},t)=\mathbf{q}+\mathbf{\psi }(\mathbf{q},t)$. Its time derivative determines the peculiar velocity through $\mathbf{v}=a\dot{\mathbf{\psi }}$.