Construction Of Variational Integrators For Lagrange Systems Based On The Local Path Fitting

Variational integrators are particularly suitable for the simulation of classical mechanical systems as the inherent geometric structure of the original continuous system is maintained.Therefore,a new method to construct the variational integrators of mechanical systems was proposed.The new method uses the discretization of variational principles and combines the continuous and discrete Euler-Lagrange equations,so that the constructed variational integrators automatically inherit the geometric characteristics of the solution of the continuous systems,which makes the constructed variational integrators very efficient in simulating the motion of the dynamic systems.Applying this new construction method to the mechanical system,it can be found that the obtained variational integrators have better performance than the variational integrators constructed by the classical construction method.According to the form of the discrete Euler-Lagrange equations,the general construction of variational integrators finally comes down to the calculation of the partial derivatives of the discrete Lagrangian,where the discrete Lagrangian is the action integral of the Lagrangian over a short time interval,usually approximated by classical quadrature rules.From the integral expression of the discrete Lagrangian,it can be seen that analytic calculation of its partial derivatives will induce a new integral closely related to the continuous Euler-Lagrange equation.Therefore,the construction of variational integrators can no longer rely on the concrete form of the discrete Lagrangian obtained through classical quadrature rules,but can directly approximate the newly arising integral based on a set of discrete nodes.If the local trajectory is further fitted by requiring the Euler-Lagrange equations hold at these interval nodes,the new integral vanishes when computed numerically.Correspondingly,the calculation of the partial derivatives of the discrete Lagrangian is simplified to the calculation of the partial derivatives of the Lagrangian with respect to the velocity variables.The new method is equivalent to the discretization of the laws of physics,so the resulting difference format is closer to the real world,and must be better than traditional algorithms in terms of stability and accuracy.When fitting the local path of the Lagrange systems,Lagrange interpolation polynomial and Bernstein interpolation polynomial are mainly used.A variety of mechanical systems such as harmonic oscillator,simple pendulum and uniform circular motion are simulated to prove the effectiveness of this new construction method,and the conclusions obtained are consistent with reality.Numerical results show that the variational integrators for Lagrange systems based on the local path fitting have better performance than the variational integrators constructed by the classical construction method.In addition,the more dense the selection of internal nodes,the more accurate the variational integrators obtained.In the initialization of simulation,the traditional Euler method is not adopted,but a new initialization method is given by applying the local path fitting and continuous Euler-Lagrange equation.We also make a detailed analysis of the error introduced by the construction method and further sum up the general conclusion.The accuracy of the obtained variational integrators is closely related to the parity of the numbers of nodes of the interpolation polynomial.