Local path fitting: A new approach to variational integrators

A new approach for constructing variational integrators is presented. In the general case, the estimation of the action integral in a time interval [tk,tk+1][tk,tk+1] is used to construct a symplectic map (qk,qk+1)→(qk+1,qk+2)(qk,qk+1)→(qk+1,qk+2). The basic idea, is that only the partial derivatives of the estimated action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, gives rise to a new integral that depends on the Euler–Lagrange vector itself (which in the continuous and exact case vanishes) and not on the Lagrangian. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the Euler–Lagrange vector to vanish at these grid points. Thus, the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested in high order solutions of the two-body problem with high eccentricity (up to 0.99) and of the outer planets of the solar system.