Two-dimensional Fireballs as a Lagrangian Ermakov System
Abstract
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 {\it 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.
Cite
@article{arxiv.2408.16521,
title = {Two-dimensional Fireballs as a Lagrangian Ermakov System},
author = {Fernando Haas},
journal= {arXiv preprint arXiv:2408.16521},
year = {2024}
}