Two-dimensional Fireballs as a Lagrangian Ermakov System

TL;DR

This paper derives and analyzes the equations governing the dynamics of two-dimensional fireballs using a Lagrangian Ermakov system, providing exact solutions and connecting symmetries to conserved quantities.

Contribution

It introduces a novel Lagrangian Ermakov system framework for two-dimensional fireballs, including exact solutions and symmetry analysis, extending previous models.

Findings

Two-dimensional fireball equations form a Lagrangian Ermakov system.

Exact analytical solutions for the fireball dynamics are obtained.

Symmetries relate to conserved quantities via Noether's theorem.

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.