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Filobello-Nino U, Vazquez-Leal H, Huerta-Chua J, Mayorga-Cruz D, Lopez-Leal R, Callejas Molina R, Sandoval-Hernandez M. A symmetric version of the Euler equations by using Generalized Bernoulli Method. Heliyon 2023; 9:e16947. [PMID: 37360111 PMCID: PMC10285128 DOI: 10.1016/j.heliyon.2023.e16947] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2021] [Revised: 08/22/2022] [Accepted: 06/01/2023] [Indexed: 06/28/2023] Open
Abstract
The aim of this article is to show a way to extend the usefulness of the Generalized Bernoulli Method (GBM) with the purpose to apply it for the case of variational problems with functionals that depend explicitly of all the variables. Moreover, after expressing the Euler equations in terms of this extension of GBM, we will see that the resulting equations acquire a symmetric form, which is not shared by the known Euler equations. We will see that this symmetry is useful because it allows us to recall these equations with ease. The presentation of three examples shows that by applying GBM, the Euler equations are obtained just as well as it does the known Euler formalism but with much less effort, which makes GBM ideal for practical applications. In fact, given a variational problem, GBM establishes the corresponding Euler equations by means of a systematic procedure, which is easy to recall, based in both elementary calculus and algebra without having to memorize the known formulas. Finally, in order to extend the practical applications of the proposed method, this work will employ GBM with the purpose to apply it for the case of solving isoperimetric problems.
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Affiliation(s)
- U. Filobello-Nino
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz, Mexico
| | - H. Vazquez-Leal
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz, Mexico
- Consejo Veracruzano de Investigación Científica y Desarrollo Tecnológico (COVEICYDET), Av Rafael Murillo Vidal No. 1735, Cuauhtémoc, 91069 Xalapa, Veracruz, Mexico
| | - J. Huerta-Chua
- Instituto Tecnológico Superior de Poza Rica, Tecnológico Nacional de México, Luis Donaldo Colosio Murrieta S/N, Arroyo del Maíz, 93230 Poza Rica, Veracruz, Mexico
| | - D. Mayorga-Cruz
- Consejo Veracruzano de Investigación Científica y Desarrollo Tecnológico (COVEICYDET), Av Rafael Murillo Vidal No. 1735, Cuauhtémoc, 91069 Xalapa, Veracruz, Mexico
- Centro de Investigación en Ingeniería y Ciencias Aplicadas, CIICAP, Universidad Autónoma del Estado de Morelos, 62209 Cuernavaca, Morelos, Mexico
| | - R. Lopez-Leal
- Consejo Veracruzano de Investigación Científica y Desarrollo Tecnológico (COVEICYDET), Av Rafael Murillo Vidal No. 1735, Cuauhtémoc, 91069 Xalapa, Veracruz, Mexico
| | - R.A. Callejas Molina
- Posgrado de Ciencias en la Ingeniería, Instituto Tecnológico de Celaya, Tecnológico Nacional de México, Antonio García Cubas Pte. 600, 38010 Celaya, Guanajuato, Mexico
| | - M.A. Sandoval-Hernandez
- CBTis 190 DGETI. Av 15, Venustiano Carranza, Carranza 2da Sección, 94297 Boca del Río, Veracruz, Mexico
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