Numerical integration of variational equations for Hamiltonian systems with long range interactions

TL;DR

This paper investigates the chaotic behavior of long-range interacting Hamiltonian lattices, revealing organized dynamics for certain interaction exponents and validating numerical methods for large systems.

Contribution

It introduces numerical techniques for integrating variational equations in long-range Hamiltonian systems and analyzes their chaotic properties across different interaction regimes.

Findings

Chaos is strong but organized behavior emerges for alpha<1.

Maximal Lyapunov exponent decreases with system size.

Numerical methods accurately conserve energy for large systems.

Abstract

We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/r^alpha, for alpha>=0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent alpha<1. This is shown by computing dynamical quantities such as the maximal Lyapunov exponent, which decreases as the number of degrees of freedom increases. We also discuss our numerical methods of symplectic integration implemented for the solution of the equations of motion together with their associated variational equations. The validity of our numerical simulations is estimated by showing that the total energy of the system is conserved within an accuracy of 4 digits (with integration step tau=0.02), even for as many as N=8000 particles and integration times as long as 10^6 units.