Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations
A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, t...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2020/3170130 |
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| author | Z. E. Musielak N. Davachi M. Rosario-Franco |
| author_facet | Z. E. Musielak N. Davachi M. Rosario-Franco |
| author_sort | Z. E. Musielak |
| collection | DOAJ |
| description | A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed. |
| format | Article |
| id | doaj-art-80ac7efc04eb4aa7a6955dd7d55f5d4d |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-80ac7efc04eb4aa7a6955dd7d55f5d4d2025-02-03T06:05:41ZengWileyJournal of Applied Mathematics1110-757X1687-00422020-01-01202010.1155/2020/31701303170130Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential EquationsZ. E. Musielak0N. Davachi1M. Rosario-Franco2Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USADepartment of Physics, The University of Texas at Arlington, Arlington, TX 76019, USADepartment of Physics, The University of Texas at Arlington, Arlington, TX 76019, USAA set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.http://dx.doi.org/10.1155/2020/3170130 |
| spellingShingle | Z. E. Musielak N. Davachi M. Rosario-Franco Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations Journal of Applied Mathematics |
| title | Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations |
| title_full | Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations |
| title_fullStr | Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations |
| title_full_unstemmed | Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations |
| title_short | Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations |
| title_sort | lagrangians gauge functions and lie groups for semigroup of second order differential equations |
| url | http://dx.doi.org/10.1155/2020/3170130 |
| work_keys_str_mv | AT zemusielak lagrangiansgaugefunctionsandliegroupsforsemigroupofsecondorderdifferentialequations AT ndavachi lagrangiansgaugefunctionsandliegroupsforsemigroupofsecondorderdifferentialequations AT mrosariofranco lagrangiansgaugefunctionsandliegroupsforsemigroupofsecondorderdifferentialequations |