# Lagrangian and Hamiltonian Taylor Variational Integrators

## TL;DR

This paper introduces a Taylor-based variational integrator that leverages the structure of Taylor's method for higher-order accuracy and efficiency in solving Euler-Lagrange boundary-value problems, with demonstrated numerical benefits.

## Contribution

It presents a novel Taylor variational integrator that achieves higher order accuracy and efficiency by exploiting Taylor's method structure, including a symmetric version.

## Key findings

- Higher order accuracy compared to existing shooting methods
- Enhanced efficiency through polynomial evaluations
- Numerical experiments confirm effectiveness

## Abstract

In this paper, we present a variational integrator that is based on an approximation of the Euler--Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.