### Abstract

We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/r^{α}, for α≥0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent α<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 τ=0.02), even for as many as N=8000 particles and integration times as long as 10^{6} units.

Original language | English |
---|---|

Pages (from-to) | 158-165 |

Number of pages | 8 |

Journal | Applied Numerical Mathematics |

Volume | 104 |

DOIs | |

Publication status | Published - Jun 1 2016 |

Externally published | Yes |

### Fingerprint

### Keywords

- Hamiltonian systems
- Long range interactions
- Symplectic integration
- Variational equations

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Numerical Mathematics*,

*104*, 158-165. https://doi.org/10.1016/j.apnum.2015.08.009

**Numerical integration of variational equations for Hamiltonian systems with long range interactions.** / Christodoulidi, Helen; Bountis, Tassos; Drossos, Lambros.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 104, pp. 158-165. https://doi.org/10.1016/j.apnum.2015.08.009

}

TY - JOUR

T1 - Numerical integration of variational equations for Hamiltonian systems with long range interactions

AU - Christodoulidi, Helen

AU - Bountis, Tassos

AU - Drossos, Lambros

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/rα, for α≥0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent α<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 τ=0.02), even for as many as N=8000 particles and integration times as long as 106 units.

AB - We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/rα, for α≥0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent α<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 τ=0.02), even for as many as N=8000 particles and integration times as long as 106 units.

KW - Hamiltonian systems

KW - Long range interactions

KW - Symplectic integration

KW - Variational equations

UR - http://www.scopus.com/inward/record.url?scp=84945567190&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945567190&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2015.08.009

DO - 10.1016/j.apnum.2015.08.009

M3 - Article

VL - 104

SP - 158

EP - 165

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -