基于Lagrange插值多项式拟合的力学系统的变分积分子

VARIATIONAL INTEGRATORS FOR MECHANICAL SYSTEMS BASED ON THE LAGRANGE INTERPOLATING POLYNOMIAL

变分积分子是通过直接离散变分原理得到的一类特殊的动力学系统的数值差分格式,较之传统差分格式呈现出明显的计算优越性.由离散Euler-Lagrange方程的形式可知,变分积分子的构造过程最终归结为计算离散Lagrange函数的偏导数,其中离散Lagrange函数是Lagrange函数在单个时间步长的积分,通常由经典求积公式近似得到.根据离散Lagrange函数的积分表达式,解析计算其偏导数会随之衍生一个新的且与连续Euler-Lagrange方程密切关联的积分,因此,构造变分积分子就可以不再以通过经典求积公式得到的具体形式的离散Lagrange函数为前提,而是可以直接基于一组离散结点近似新衍生的积分.在这些离散结点处,如果进一步让系统的拟合轨迹严格满足Euler-Lagrange方程,即运动方程,那么新的积分自动为零,相应地,计算离散Lagrange函数的偏导数就简化为计算连续Lagrange函数关于速度变量的偏导数.这种新的构造方式同时结合了连续和离散的Euler-Lagrange方程,不仅让最终得到的差分格式仍然继承了变分积分子特有的优越计算性能,而且在同阶精度的情况下具有更小的局部误差.

Variational integrators are a special kind of numerical difference schemes for mechanical systems that are de⁃rived from discrete variational principles.They exhibit obvious superiority to classical numerical algorithms.As obvious⁃ly shown in the discrete Euler-Lagrange equations,the general construction of variational integrators finally comes down to the calculation of partial derivatives of the discrete Lagrangian,which is an approximation of the action integral of the Lagrangian over a short time interval.Inspired by this fact,analytic calculation of these partial derivatives is carried out,which induces a new integral depending on the Euler-Lagrange equations.Therefore,the evaluation of the integral of the Lagrangian by using any quadrature rule can be transformed into the evaluation of the newly induced integral based on a series of interval nodes.If the local trajectory is further fitted by requiring that the Euler-Lagrange equations hold at these interval nodes,the new integral will vanish when being computed through any numerical method.And according⁃ly,the calculation of the partial derivative of the discrete Lagrangian is simplified to calculate the partial derivative of the Lagrangian with respect to the velocity.The resulting algorithms of the new approach,which combine both the contin⁃uous and discrete Euler-Lagrange equations,not only preserve those unique benefits of variational integrators,but also produce smaller local errors with similar accuracy being attained.