Local Path Fitting: A New Approach to Variational Integrators

TL;DR

This paper introduces Local Path Fitting, a novel method for constructing variational integrators that simplifies high-order approximations by locally fitting the exact solution curve using the Euler--Lagrange vector at grid points.

Contribution

It proposes a new approach to variational integrators that relies on local path fitting and the Euler--Lagrange vector, reducing computational complexity for high-order solutions.

Findings

Successfully applied to high eccentricity two-body problem

Achieved high-order solutions in outer solar system simulations

Simplified calculation of variational integrators

Abstract

In this work, we present a new approach to the construction of variational integrators. In the general case, the estimation of the action integral in a time interval $[q_k,q_{k+1}]$ is used to construct a symplectic map $(q_k,q_{k+1})\to (q_{k+1},q_{k+2})$. The basic idea here, is that only the partial derivatives of the estimation of the action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, give raise to a new integral which depends not on the Lagrangian but on the Euler--Lagrange vector, which in the continuous and exact case vanishes. 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 for high order solutions in the two-body problem with high eccentricity (up to 0.99) and in the outer solar system.