MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE
In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the indepe...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2016-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509416000244/type/journal_article |
| _version_ | 1811156229620236288 |
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| author | TÂNIA M. N. GONÇALVES ELIZABETH L. MANSFIELD |
| author_facet | TÂNIA M. N. GONÇALVES ELIZABETH L. MANSFIELD |
| author_sort | TÂNIA M. N. GONÇALVES |
| collection | DOAJ |
| description | In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables. |
| first_indexed | 2024-04-10T04:47:03Z |
| format | Article |
| id | doaj.art-48f31c9f15a749b8874f3f701924f21c |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:03Z |
| publishDate | 2016-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-48f31c9f15a749b8874f3f701924f21c2023-03-09T12:34:41ZengCambridge University PressForum of Mathematics, Sigma2050-50942016-01-01410.1017/fms.2016.24MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASETÂNIA M. N. GONÇALVES0ELIZABETH L. MANSFIELD1Unidade Acadêmica Especial de Matemática e Tecnologia, Universidade Federal de Goiás, Catalão 75704-020, Brazil;School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK;In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.https://www.cambridge.org/core/product/identifier/S2050509416000244/type/journal_article58E3022E70 (primary)34A2634K17 (secondary) |
| spellingShingle | TÂNIA M. N. GONÇALVES ELIZABETH L. MANSFIELD MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE Forum of Mathematics, Sigma 58E30 22E70 (primary) 34A26 34K17 (secondary) |
| title | MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE |
| title_full | MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE |
| title_fullStr | MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE |
| title_full_unstemmed | MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE |
| title_short | MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE |
| title_sort | moving frames and noether s conservation laws the general case |
| topic | 58E30 22E70 (primary) 34A26 34K17 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509416000244/type/journal_article |
| work_keys_str_mv | AT taniamngoncalves movingframesandnoethersconservationlawsthegeneralcase AT elizabethlmansfield movingframesandnoethersconservationlawsthegeneralcase |