Laméris, C.L. (2020) (Non-)integrability of Hamiltonian systems via differential Galois theory and the Painlevé property. Bachelor's Thesis, Mathematics.
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Abstract
In this report we shall introduce differential Galois theory. To do so rigorously, we shall also discuss preliminaries from algebraic geometry and algebraic groups. We shall see how differential Galois theory can be used to show whether a differential equation is integrable or not, the ultimate purpose to apply differential Galois theory to the non-integrability of Hamiltonian systems. We do so via so-called Ziglin-Morales-Ramis theory, and apply the theory to completely determine the (non-)integrability of the spring pendulum. We shall show what it means for a Hamiltonian system to be completely integrable using Hamlitonian formalism, and what it means when a systems is not completely integrable, via the KAM and Nekhorosev theorems. We shall also discuss the Painleé equations and property, and show how this property relates to the complete integrability of Hamiltonian systems. As an example we find integrable cases of the Hénon-Heiles system using the (ARS) Painlevé test.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Top, J. and Noordman, M.P. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 06 Aug 2020 08:24 |
| Last Modified: | 24 Aug 2020 13:29 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/22836 |
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