Gunnink, Fabian (2026) Symmetry algebras of Bertrand Hamiltonians. Master's Thesis / Essay, Physics.
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Abstract
The Kepler-Coulomb and harmonic oscillator systems occupy a privileged position in classical mechanics: beyond their rotational symmetry, both possess hidden constants of motion. Their additional structure elevates the symmetry algebra from so(3) to so(4) and su(3) respectively, and is responsible for the closure of all bounded orbits, a result known as Bertrand's theorem. This thesis investigates whether this exceptional algebraic structure survives once the geometry of the ambient space is no longer flat. In this way, two classes of Hamiltonian systems arise. Class-I Hamiltonians have two deformation parameters, β and K, while those in class-II(±) have an additional parameter D. We show that the class-I and class-II(+) (with D = 0) Hamiltonians are nothing but the Kepler-Coulomb and harmonic oscillator systems on the spaces of constant curvature, respectively. With the construction of a curved Laplace-Runge-Lenz vector and Fradkin-Demkov tensor, we show that the symmetry algebras remain, respectively, so(4) and su(3) in these cases. In the general class-II(+) case with K, D ≠ 0, the metric has a non-constant curvature. For these cases, we propose a deformed Fradkin-Demkov tensor that generates a cubic Poisson algebra, with the underlying geometry identified as the Darboux-IV space. These results demonstrate that the hidden symmetry algebras so(4) and su(3) are stable features of a much larger set of Hamiltonian systems, rather than two mere accidents in Euclidean geometry.
| Item Type: | Thesis (Master's Thesis / Essay) |
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| Supervisor name: | Roest, D. and Seri, M. |
| Degree programme: | Physics |
| Thesis type: | Master's Thesis / Essay |
| Language: | English |
| Date Deposited: | 02 Jun 2026 13:24 |
| Last Modified: | 02 Jun 2026 13:24 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/37312 |
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