In this paper we provide a variational derivation of the Euler–Poincare equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others Martin de Diego and Martin de Almagro (2018 Nonlinearity 31 3814–3846), Galley (2013 Phys. Rev. Lett. 110 174301), Galley et al (2014 (arXiv:[math-Ph] 1412.3082)). Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the previous construction to the formal derivation of the variational error for numerical integrators of forced Euler–Poincare equations and the application of this theory to the derivation of geometric integrators for forced systems.

中文翻译：

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强制拉格朗日系统的变分阶 II。带强迫的欧拉-庞加莱方程

在本文中，我们使用由 Galley 和其他人 Martin de Diego 和 Martin de Almagro (2018 Nonlinearity 31 3814–3846)、Galley (2013) 引入的技术，提供了对受到外力的系统的欧拉-庞加莱方程的变分推导Phys. Rev. Lett. 110 174301)，Galley 等人 (2014 (arXiv:[math-Ph] 1412.3082))。此外，我们详细研究了与 Poisson groupoid 概念相关的基础几何。最后，我们将先前的构造应用于强制 Euler-Poincare 方程的数值积分器的变分误差的形式推导，并将该理论应用于强制系统的几何积分器的推导。