The nonlinear pendulum: formulas for the large amplitude period
P. Amore*, M. Cervantes Valdovinos, G. Ornelas, and S. Zamudio Barajas
Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, México, e–mail: paolo@ucol.mx.
Recibido el 10 de agosto de 2006
Aceptado el 11 de septiembre de 2006
Abstract
A simple and precise formula for the period of a nonlinear pendulum is obtained using the Linear Delta Expansion, a powerful non–perturbative technique which has been applied in the past to problems in different areas of physics. Our result is based on a systematic approach which allows us to obtain a new series for the elliptic integrals, in terms of which the exact solution of our problem is cast. A further improvement of the LDE result is then obtained by using Pade approximants. Finally we make a comparison with other approximations in the literature for the period of the pendulum, valid either at small or at large angles.
Keywords: Perturbation theory; linear delta expansion; pendulum.
Resumen
Hemos obtenido una fórmula simple y precisa para el período de oscilación de un péndulo no–lineal utilizando la Expansión Delta Lineal, una técnica no–perturbativa que ha sido utilizada en el pasado en áreas distintas de la física. Nuestro resultado se ha obtenido utilizando un método sistemático que permite obtener una serie para las integrales elípticas, por medio de la cual se expresa la solución exacta. Los resultados son mejorados también utilizando los aproximantes de Padé. Finalmente, hacemos también una comparación con otras aproximaciones en la literatura para el periodo del péndulo, tanto para ángulos pequeños como para grandes.
Descriptores: Teoría de perturbaciones; expansión delta lineal; péndulo.
PACS: 04.25.–0; 4.20.Jbg; 01.55.+b
]]>DESCARGAR ARTÍCULO EN FORMATO PDF
Acknowledgments
P.A. acknowledges support of Conacyt grant no. C01–40633/A–1. P.A. and G.O. also acknowledge support of Fondo Ramón Alvarez Buylla of Colima University.
References
1. W.P. Ganley, Am. J. Phys. 53 (1995) 73. [ Links ]
2. R.B. Kidd and S.L. Fogg, Phys. Teach. 40 (2003) 81. [ Links ]
3. M.I. Molina, Phys. Teach. 35 (1997) 489. [ Links ]
4. P. Amore and R.A. Sáenz, Europhysics Letters 70 (2005) 425. [ Links ]
5. B. Wu and P. Li, Meccanica 36 (2001) 167. [ Links ]
6. R. Parwani, European Journal of Physics 25 (2004) 37. [ Links ]
7. A. Cromer, American Journal of Physics 63 (1995) 112. [ Links ]
8. L.M. Burko, European Journal of Physics 24 (2002) 125. [ Links ]
9. A. Duncana and M. Moshe, Phys. Lett. B 125 (1998) 352; [ Links ] A.Okopinska, Phys. Rev. D 35 (1987) 1835; [ Links ] G. Krein, D.P Menezes, and M.B. Pinto, Phys. Lett. B 370 (1996) 5; [ Links ] M.P Blencowe and A.P Korte, Phys. Rev. B 56 (1997) 9422; [ Links ] H.F. Jones, P. Parkin, and D. Winder, Phys. Rev. D 63 (2001) 125013; [ Links ] J.L. Kneur, M.B. Pinto, and R.O. Ramos, Phys. Rev. Lett. 89 (2002) 210403; [ Links ] A. Pelster, H. Kleinert, and M. Schanz, Phys. Rev. E 67 (2003) 016604. [ Links ]
10. G.A. Arteca, F.M. Fernández, and E.A. Castro, Large order perturbation theory and summation methods in quantum mechanics (Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, 1990). [ Links ]
11. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, 3rd edition (World Scientific Publishing, 2004). [ Links ]
12. P.M. Stevenson, Phys. Rev. D 23 (1981) 2916. [ Links ]
13. Handbook of Mathematical Functions, edited by M. Abramowitz and I.A. Stegun (Dover, New York, 1965). [ Links ] ]]>