Non-integrability criterium for normal variational equations aroundan integrable subsystem and an example: The Wilberforcespring-pendulum

Primitivo B. Acosta-Humánez; Martha Alvarez-Ramírez; David Blázquez-Sanz; Joaquín Delgado

doi:10.3934/dcds.2013.33.965

Article Contents

2013, Volume 33, Issue 3: 965-986. Doi: 10.3934/dcds.2013.33.965

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Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

1.
Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla
2.
Departamento de Matemáticas, UAM-Iztapalapa, 09340 Iztapalapa, México, D.F.
3.
Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C.
4.
Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, 09340 Iztapalapa, México, D. F.

Received: March 31, 2011

Revised: February 29, 2012

Published: February 2013

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Abstract

In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.

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Mathematics Subject Classification: Primary: 37J30; Secondary: 12H05 34M45 37J35 70G55 70H06.

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References

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