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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.16062 |
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| _version_ | 1866917557067317248 |
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| author | Struckmeier, Jürgen Redelbach, Andreas |
| author_facet | Struckmeier, Jürgen Redelbach, Andreas |
| contents | A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the `energy-second-moment map' that can be constructed for all Hamiltonian systems of the generic form $H=p^{2}/2+V(q,t)$. With a three-component vector $s$ consisting of the system's energy $H$ and second moments $qp$, $q^{2}$, this map linearly relates the vector $s(t)$ at time $t$ with the vector's initial state $s(0)$ at $t=0$. It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Energy-second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems Struckmeier, Jürgen Redelbach, Andreas Computational Physics Dynamical Systems A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the `energy-second-moment map' that can be constructed for all Hamiltonian systems of the generic form $H=p^{2}/2+V(q,t)$. With a three-component vector $s$ consisting of the system's energy $H$ and second moments $qp$, $q^{2}$, this map linearly relates the vector $s(t)$ at time $t$ with the vector's initial state $s(0)$ at $t=0$. It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples. |
| title | Energy-second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems |
| topic | Computational Physics Dynamical Systems |
| url | https://arxiv.org/abs/2303.16062 |