Symplectic schemes applied to Hamiltonian systems have prominent advantages for the preservation of qualitative properties of the flow. Three types of symplectic methods, which contain the symplectic Euler, implicit midpoint and Störmer–Verlet methods, are simplest and widely used in actual calculations. In this paper, we introduce a simple symplectic scheme with three free parameters, which covers these three methods and has the same behaviors as them. The symplecticity of this scheme is verified from partitioned Runge–Kutta methods and variational integrators. In addition, we get a second-order symplectic scheme with two free parameters and a symmetric symplectic scheme with a free parameter. By adjusting the parameter at each time step, we get a second-order energy and quadratic invariants preserving method. The effectiveness of all schemes is demonstrated by numerical tests.

适用于哈密顿系统的辛格式在保持流的定性方面具有显着优势。三种辛方法，其中包括辛欧拉方法，隐中点方法和Störmer-Verlet方法，是最简单的方法，并且在实际计算中广泛使用。在本文中，我们介绍了一个带有三个自由参数的简单辛格方案，它涵盖了这三种方法，并且具有与它们相同的行为。通过划分的Runge–Kutta方法和变分积分器验证了该方案的辛性。另外，我们得到了带有两个自由参数的二阶辛格式和一个带有自由参数的对称辛格式。通过在每个时间步长调整参数，我们得到了二阶能量和二次不变性的保存方法。