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中文摘要：
提出了哈密顿动力系统的一个新变分原理，并基于此变分原理构造了四类保辛算法。通过新的变分原理定义修正作用量，然后将位移和动量采用拉格朗日多项式近似，并采用高斯积分对时间近似积分得到近似的修正作用量。在修正作用量的基础上，通过选择时间步两端不同的位移或动量作为独立变量，可构造四种不同类型的保辛算法。
英文摘要：
In this paper, a new variational principle is proposed for the finite dimensional autonomous Hamiltonian systems and four types of symplecitc numerical algorithms are constructed. A modified ac- tion is defined by using the new variational principle. Then, the approximated action is obtained by approximating the generalized coordinates and momenta by Lagrange polynomials within a time step, and approximating the time integrals by means of Gaussian quadrature. Based on the approximated action and by taking generalized coordinates or momenta as independent variables at each end of the time step, four types of symplecitc numerical algorithms are constructed. In this paper, the detailed procedure for the construction of the algorithms is given and the proof of symplectic property and the numerical perform- ance of the algorithms will present in other papers.
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