We consider a Lagrangian system $L(q,\dot q) = \sum_{l=1}^{N}L^{\{l\}}(q,\dot q)$, where the $q$-variable is treated by a Generalized Additive Runge--Kutta (GARK) method. Applying the technique of discrete variations, we show how to construct symplectic schemes. Assuming the diagonal methods for the GARK given, we present some techinques for constructing the transition matrices. We address the problem of the order of the methods and discuss some semi-separable and separable problems, showing some interesting constructions of methods with non-square coefficient matrices.

中文翻译：

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离散变分法和辛广义加法龙格--Kutta方法

我们考虑一个拉格朗日系统 $L(q,\dot q) = \sum_{l=1}^{N}L^{\{l\}}(q,\dot q)$，其中 $q$-变量由广义可加 Runge--Kutta (GARK) 方法处理。应用离散变化的技术，我们展示了如何构造辛方案。假设给定 GARK 的对角线方法，我们提出了一些构造过渡矩阵的技术。我们解决了方法的顺序问题并讨论了一些半可分离和可分离的问题，展示了一些具有非平方系数矩阵的方法的有趣构造。