We propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler-Lagrange equations of a variational principle. The method is based on the invariantization of standard, non-invariant discrete Lagrangian functionals using equivariant moving frames. The invariant variational schemes are given by the Euler-Lagrange equations of the corresponding invariantized discrete Lagrangian functionals. We showcase this general method by constructing invariant variational schemes of ordinary differential equations that preserve variational and divergence symmetries of the associated continuous Lagrangians. Noether's theorem automatically implies that the resulting schemes are exactly conservative. Numerical simulations are carried out and show that these invariant variational schemes outperform standard numerical discretizations.

翻译：我们提出一种新的算法方法,用于构建普通差异方程式的无变式变换办法,即变式原则的Euler-Lagrange方程式。该方法基于使用等式移动框架的标准、非异离性Lagrangian功能的变换,由相应的变异离性Lagrangian函数的Euler-Lagrange方程式提供。我们通过构建普通差异方程式的变换办法展示了这一通用方法,这些变异办法维护了连续连续的Lagrangians的变异和差异对称。Noether的理论自动意味着由此产生的方案非常保守。进行了数值模拟,并表明这些变异性变异方案超越了标准数字离异化。