Invariants for single-component plasmas in a time-dependent harmonic trap.

We investigate the complex nonlinear dynamics of a quasi-one-dimensional single-component plasma confined in a time-dependent harmonic trap. The system is described as a fluid in a hydrodynamic framework, incorporating electrostatic and thermal effects. By applying a time-dependent variational method, with a Gaussian ansatz for the number density distribution, we reduce the plasma complexity to that of a system of ordinary differential equations and follow up with a systematic analysis of the corresponding Noether symmetries in three distinct regimes: electrostatic, thermal, and combined effects. For each studied case, proper conserved quantities are identified, including a generalized Ermakov-Lewis invariant. We illustrate our results in the antiproton and positron confinement problems.