Noether’s Theorem and Symmetry
In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s),...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/10/12/744 |
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| author | Amlan K. Halder Andronikos Paliathanasis Peter G.L. Leach |
| author_facet | Amlan K. Halder Andronikos Paliathanasis Peter G.L. Leach |
| author_sort | Amlan K. Halder |
| collection | DOAJ |
| description | In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables. |
| first_indexed | 2024-04-11T18:44:12Z |
| format | Article |
| id | doaj.art-698759ed70ba47688462e38dac853e83 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T18:44:12Z |
| publishDate | 2018-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-698759ed70ba47688462e38dac853e832022-12-22T04:08:54ZengMDPI AGSymmetry2073-89942018-12-01101274410.3390/sym10120744sym10120744Noether’s Theorem and SymmetryAmlan K. Halder0Andronikos Paliathanasis1Peter G.L. Leach2Department of Mathematics, Pondicherry University, Kalapet 605014, IndiaInstituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, ChileInstitute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South AfricaIn Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables.https://www.mdpi.com/2073-8994/10/12/744Noether’s theoremaction integralgeneralized symmetryfirst integralinvariantnonlocal transformationboundary termconservation lawsanalytic mechanics |
| spellingShingle | Amlan K. Halder Andronikos Paliathanasis Peter G.L. Leach Noether’s Theorem and Symmetry Symmetry Noether’s theorem action integral generalized symmetry first integral invariant nonlocal transformation boundary term conservation laws analytic mechanics |
| title | Noether’s Theorem and Symmetry |
| title_full | Noether’s Theorem and Symmetry |
| title_fullStr | Noether’s Theorem and Symmetry |
| title_full_unstemmed | Noether’s Theorem and Symmetry |
| title_short | Noether’s Theorem and Symmetry |
| title_sort | noether s theorem and symmetry |
| topic | Noether’s theorem action integral generalized symmetry first integral invariant nonlocal transformation boundary term conservation laws analytic mechanics |
| url | https://www.mdpi.com/2073-8994/10/12/744 |
| work_keys_str_mv | AT amlankhalder noetherstheoremandsymmetry AT andronikospaliathanasis noetherstheoremandsymmetry AT peterglleach noetherstheoremandsymmetry |