Two-dimensional fireballs as a Lagrangian Ermakov system

The equations of motion for the variance of strictly one-dimensional or two-dimensional non-relativistic fireballs are derived, from the hydrodynamic equations for an ideal, structureless Boltzmann gas. For this purpose a Gaussian number density Ansatz is applied, together with low-dimensional proposals for the energy density, coherent with the equipartition theorem. The resulting ordinary differential equations are shown to admit a variational formulation. The underlying symmetries are connected to constants of motion, through Noether's theorem. The two-dimensional case is special, corresponding to a Lagrangian Ermakov system without external forcing. There is a comparison with the fully three-dimensional fireballs, and its reduction to effective two-dimensional dynamical system for elliptic trajectories. The exact analytical solutions are worked out. •The equations of motion for the variance of non-relativistic fireballs in arbitrary dimensions are derived.•For this purpose a Gaussian number density Ansatz is applied, together with an adequate proposal for the energy density.•The symmetries of the action functional are connected to constants of motion, through Noether's theorem.•The two-dimensional case is an exactly integrable Lagrangian Ermakov system.•Exact analytic solutions are worked out for arbitrary dimensionality.