Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges
The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group <i>G</i>, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions...
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| Format: | Article |
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MDPI AG
2022-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/4/661 |
| _version_ | 1797443529335635968 |
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| author | Víctor Ayala Adriano Da Silva José Ayala |
| author_facet | Víctor Ayala Adriano Da Silva José Ayala |
| author_sort | Víctor Ayala |
| collection | DOAJ |
| description | The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group <i>G</i>, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on <i>G</i>: linear and invariant vector fields. A linear vector field on <i>G</i> is determined by its flow, a 1-parameter group of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the Lie group of <i>G</i>-automorphisms. An invariant vector field is just an element of the Lie algebra g of <i>G</i>. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and <i>G</i>. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper. |
| first_indexed | 2024-03-09T12:57:23Z |
| format | Article |
| id | doaj.art-eec45d742dfe461b8ec9b2334e09c8a8 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:57:23Z |
| publishDate | 2022-03-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-eec45d742dfe461b8ec9b2334e09c8a82023-11-30T21:58:41ZengMDPI AGSymmetry2073-89942022-03-0114466110.3390/sym14040661Geometric Structures Generated by the Same Dynamics. Recent Results and ChallengesVíctor Ayala0Adriano Da Silva1José Ayala2Instituto de Alta Investigación, Universidad de Tarapacá, Arica 1000000, ChileInstituto de Matemática, Universidade Estadual de Campinas, Campinas 13083-872, BrazilFacultad de Ciencias, Universidad Arturo Prat, Iquique 1100000, ChileThe main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group <i>G</i>, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on <i>G</i>: linear and invariant vector fields. A linear vector field on <i>G</i> is determined by its flow, a 1-parameter group of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the Lie group of <i>G</i>-automorphisms. An invariant vector field is just an element of the Lie algebra g of <i>G</i>. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and <i>G</i>. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper.https://www.mdpi.com/2073-8994/14/4/661linear control systemsalmost Riemannian structuresdegenerate dynamical systemssingular locus |
| spellingShingle | Víctor Ayala Adriano Da Silva José Ayala Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges Symmetry linear control systems almost Riemannian structures degenerate dynamical systems singular locus |
| title | Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges |
| title_full | Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges |
| title_fullStr | Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges |
| title_full_unstemmed | Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges |
| title_short | Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges |
| title_sort | geometric structures generated by the same dynamics recent results and challenges |
| topic | linear control systems almost Riemannian structures degenerate dynamical systems singular locus |
| url | https://www.mdpi.com/2073-8994/14/4/661 |
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