Runge-Kutta methods are affine equivariant: applying a method before or after an affine change of variables yields the same numerical trajectory. However, for some applications, one would like to perform numerical integration after a quadratic change of variables. For example, in Lie-Poisson reduction, a quadratic transformation reduces the number of variables in a Hamiltonian system, yielding a more efficient representation of the dynamics. Unfortunately, directly applying a symplectic Runge-Kutta method to the reduced system generally does not preserve its Hamiltonian structure, so many proposed techniques require computing numerical trajectories of the original, unreduced system. In this paper, we study when a Runge-Kutta method in the original variables descends to a numerical integrator expressible entirely in terms of the quadratically transformed variables. In particular, we show that symplectic diagonally implicit Runge-Kutta (SyDIRK) methods, applied to a quadratic projectable vector field, are precisely the Runge-Kutta methods that descend to a method (generally not of Runge-Kutta type) in the projected variables. We illustrate our results with several examples in both conservative and non-conservative dynamics.

翻译：龙格-库塔方法具有仿射等变性：在变量进行仿射变换前后应用该方法会得到相同的数值轨迹。然而，在某些应用中，人们希望在变量进行二次变换后进行数值积分。例如，在李-泊松约化中，二次变换可减少哈密顿系统中的变量数目，从而得到更高效的动力学表示。遗憾的是，直接将辛龙格-库塔方法应用于约化系统通常无法保持其哈密顿结构，因此许多现有技术需要计算原始未约化系统的数值轨迹。本文研究了当原始变量中的龙格-库塔方法可降阶为完全用二次变换变量表示的数值积分器的条件。特别地，我们证明了应用于二次可投影向量场的辛对角隐式龙格-库塔方法，正是那些能降阶为投影变量中数值方法（通常非龙格-库塔型）的龙格-库塔方法。我们通过保守与非保守动力学中的若干算例验证了所得结论。