Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws

We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor $g_{\mu\nu}(\bm\xi,t)$ and a skew-symmetric vorticity tensor $\widetilde{F}_{\mu\nu}(\bm\xi,t)$, where $\bm\xi$ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric $g_{\mu\nu}(\bm\xi,t)$ in the presence of an effective magnetic field $\widetilde{F}_{\mu\nu}(\bm\xi,t)$. To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.