The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems
In this paper, we analyse the classical action as a tool to reveal the phase space structure of Hamiltonian systems simply and intuitively. We construct a scalar field using the values of the action along the trajectories to analyse the phase space. The different behaviours of the trajectories aroun...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Dynamics |
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| Online Access: | https://www.mdpi.com/2673-8716/3/4/36 |
| _version_ | 1797381380950196224 |
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| author | Francisco Gonzalez Montoya |
| author_facet | Francisco Gonzalez Montoya |
| author_sort | Francisco Gonzalez Montoya |
| collection | DOAJ |
| description | In this paper, we analyse the classical action as a tool to reveal the phase space structure of Hamiltonian systems simply and intuitively. We construct a scalar field using the values of the action along the trajectories to analyse the phase space. The different behaviours of the trajectories around important geometrical objects like normally hyperbolic invariant manifolds, their stable and unstable manifolds, and KAM structures generate characteristic patterns in the scalar field generated by the action. Also, we present a simple argument based on the conservation of energy and the behaviour of the trajectories to understand the origin of the patterns in this scalar field. As examples, we study the phase space of open Hamiltonian systems with two and three degrees of freedom. |
| first_indexed | 2024-03-08T20:50:39Z |
| format | Article |
| id | doaj.art-4ed1e6f8fc1d4505a3e5dcc27dd50df9 |
| institution | Directory Open Access Journal |
| issn | 2673-8716 |
| language | English |
| last_indexed | 2024-03-08T20:50:39Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Dynamics |
| spelling | doaj.art-4ed1e6f8fc1d4505a3e5dcc27dd50df92023-12-22T14:04:02ZengMDPI AGDynamics2673-87162023-10-013467869410.3390/dynamics3040036The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian SystemsFrancisco Gonzalez Montoya0Faculty of Physical Sciences and Engineering, University of Leeds, Leeds LS2 9JT, UKIn this paper, we analyse the classical action as a tool to reveal the phase space structure of Hamiltonian systems simply and intuitively. We construct a scalar field using the values of the action along the trajectories to analyse the phase space. The different behaviours of the trajectories around important geometrical objects like normally hyperbolic invariant manifolds, their stable and unstable manifolds, and KAM structures generate characteristic patterns in the scalar field generated by the action. Also, we present a simple argument based on the conservation of energy and the behaviour of the trajectories to understand the origin of the patterns in this scalar field. As examples, we study the phase space of open Hamiltonian systems with two and three degrees of freedom.https://www.mdpi.com/2673-8716/3/4/36Hamiltonian systemsphase space analysisinvariant manifolds |
| spellingShingle | Francisco Gonzalez Montoya The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems Dynamics Hamiltonian systems phase space analysis invariant manifolds |
| title | The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems |
| title_full | The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems |
| title_fullStr | The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems |
| title_full_unstemmed | The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems |
| title_short | The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems |
| title_sort | classical action as a tool to visualise the phase space of hamiltonian systems |
| topic | Hamiltonian systems phase space analysis invariant manifolds |
| url | https://www.mdpi.com/2673-8716/3/4/36 |
| work_keys_str_mv | AT franciscogonzalezmontoya theclassicalactionasatooltovisualisethephasespaceofhamiltoniansystems AT franciscogonzalezmontoya classicalactionasatooltovisualisethephasespaceofhamiltoniansystems |