Second-order variational equations for N-body simulations

TL;DR

This paper extends variational equations in N-body simulations to second order, enabling advanced optimization and sampling methods with improved convergence and efficiency, and provides an implementation in the REBOUND package.

Contribution

It derives second-order variational equations for N-body problems and integrates them into the REBOUND software, enhancing capabilities for optimization and statistical sampling.

Findings

Derived second-order variational equations for N-body systems.

Implemented the equations in the REBOUND integrator package.

Enabled simultaneous integration of multiple variational equations.

Abstract

First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton's method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection. We provide an implementation of first and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.