Two-dimensional Fireballs as a Lagrangian Ermakov System
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929478583713792 |
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| author | Haas, Fernando |
| author_facet | Haas, Fernando |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-dimensional Fireballs as a Lagrangian Ermakov System Haas, Fernando Mathematical Physics 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. |
| title | Two-dimensional Fireballs as a Lagrangian Ermakov System |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2408.16521 |