# A Symplectic Integrator for Hill's Equations

## TL;DR

This paper introduces a symplectic integrator based on a generalized leapfrog method for Hill's equations, enhancing long-term orbital simulations in astrophysics with improved accuracy and efficiency.

## Contribution

A novel symplectic integrator tailored for Hill's equations, implemented in PKDGRAV, enabling stable long-term integrations and efficient collision detection.

## Key findings

- Demonstrates minimal secular drift in orbital elements over many dynamical times
- Efficient collision searching via linear extrapolation of particle positions
- Validated on simple orbital scenarios

## Abstract

Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.