We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state and we approximate the action in a small time step by the Simpson's quadrature formula. The resulting scheme is explicited for an elementary harmonic oscillator. It is a stable, explicit, and symplectic scheme satisfying the conservation of an approximate energy. Numerical tests illustrate our theoretical study.

翻译：本文提出了一种基于最小作用量原理的动力学系统时间积分的变分辛数值方法。我们假设系统状态具有二次内部插值特性，并采用辛普森求积公式在小时步内逼近作用量。以基本谐振子为例推导了具体计算格式，该格式具有稳定、显式和辛几何特性，且满足近似能量守恒。数值实验验证了理论研究的有效性。