The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry
Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differen...
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| Format: | Article |
| Language: | Spanish |
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Universidad Nacional de Trujillo
2024-12-01
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| Series: | Selecciones Matemáticas |
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| Online Access: | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6163 |
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| author | Delphin Mwinken |
| author_facet | Delphin Mwinken |
| author_sort | Delphin Mwinken |
| collection | DOAJ |
| description | Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length functionals. This paper explores the intrinsic connections between these areas, highlighting key principles such as the Euler-Lagrange equation, Ekeland’s variational principle, and the Mountain Pass theorem, and their applications in solving PDEs and understanding geometrical structures. |
| format | Article |
| id | doaj-art-1ff235d9e4cc4e74b2071e66ba6d70fc |
| institution | Kabale University |
| issn | 2411-1783 |
| language | Spanish |
| publishDate | 2024-12-01 |
| publisher | Universidad Nacional de Trujillo |
| record_format | Article |
| series | Selecciones Matemáticas |
| spelling | doaj-art-1ff235d9e4cc4e74b2071e66ba6d70fc2025-01-03T03:56:36ZspaUniversidad Nacional de TrujilloSelecciones Matemáticas2411-17832024-12-01110239340810.17268/sel.mat.2024.02.11The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential GeometryDelphin Mwinken0https://orcid.org/0000-0002-2540-9027Obuda University Doctoral School of Applied Informatics and Applied Mathematics Budapest-Hungary. High Polytechnics Institute of Jos´e Edurado University-Huambo-AngolaCalculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length functionals. This paper explores the intrinsic connections between these areas, highlighting key principles such as the Euler-Lagrange equation, Ekeland’s variational principle, and the Mountain Pass theorem, and their applications in solving PDEs and understanding geometrical structures.https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6163calculus of variationsfunctional optimizationpartial differential equationsdifferential geometryeuler-lagrange equationekeland’s variational principlemountain pass theoremgeodesics |
| spellingShingle | Delphin Mwinken The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry Selecciones Matemáticas calculus of variations functional optimization partial differential equations differential geometry euler-lagrange equation ekeland’s variational principle mountain pass theorem geodesics |
| title | The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry |
| title_full | The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry |
| title_fullStr | The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry |
| title_full_unstemmed | The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry |
| title_short | The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry |
| title_sort | interconnection between calculus of variations partial differential equations and differential geometry |
| topic | calculus of variations functional optimization partial differential equations differential geometry euler-lagrange equation ekeland’s variational principle mountain pass theorem geodesics |
| url | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6163 |
| work_keys_str_mv | AT delphinmwinken theinterconnectionbetweencalculusofvariationspartialdifferentialequationsanddifferentialgeometry AT delphinmwinken interconnectionbetweencalculusofvariationspartialdifferentialequationsanddifferentialgeometry |