Symplectic integrator

From Wikipedia, the free encyclopedia

In mathematics, a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related to classical mechanics. Symplectic integrators are a class of geometric integrators. They are widely used in molecular dynamics, discrete element methods and celestial mechanics.

[edit] Introduction

Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read

$\dot p = -\frac{\partial H}{\partial q} \quad\mbox{and}\quad \dot q = \frac{\partial H}{\partial p},$

where $q$ denotes the position coordinates, $p$ the momentum coordinates, and $H$ is the Hamiltonian (see Hamiltonian mechanics for more background).

The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic two-form $dp \wedge dq$ . A numerical scheme is a symplectic integrator if it also conserves this two-form.

Symplectic integrators possess as a conserved quantity a Hamiltonian which is slightly perturbed from the original one. By virtue of these advantages, the SI scheme has been widely applied to the calculations of long-term evolution of chaotic Hamiltonian systems ranging from the Kepler problem to the classical and semi-classical simulations in molecular dynamics.

Most of the usual numerical methods, like the primitive Euler scheme and the classical Runge-Kutta scheme, are not symplectic integrators.

[edit] Splitting methods for separable Hamiltonians

A widely used class of symplectic integrators is formed by the splitting methods.

Assume that the Hamiltonian is separable, meaning that it can be written in the form

$H(p,q) = T(p) + V(q). \qquad\qquad (1)$

This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy.

Then the equations of motion of a Hamiltonian system can be expressed as

$\dot{z}=\{z,H(z)\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

where $\{\cdot, \cdot\}$ is a Poisson bracket. By using the notation $D_H = \{\cdot, H\}$ , this can be re-expressed as

$\dot{z}=D_H z.$

The formal solution of this set of equations is given as

$z(\tau)=\exp(\tau D_H)z(0). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

When the Hamiltonian has the form of eq. (1), the solution (3) is equivalent to

$z(\tau) = \exp[\tau (D_T + D_V)]z(0). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

The SI scheme approximates the time-evolution operator $exp[τ(D T + D V)]$ in the formal solution (4) by a product of operators as

$\exp[\tau (D_T + D_V)] = \Pi_{i=1}^k \exp(c_i \tau D_T)\exp(d_i \tau D_V) + O(\tau^{n+1}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$

where $c i$ and $d i$ are real numbers, and $n$ is an integer, which is called the order of the integrator. Note that each of the operators $exp(c i τ D T)$ and $exp(d i τ D V)$ provides a symplectic map, so their product appearing in the right hand side of (5) also constitutes a symplectic map. In concrete terms, $exp(c i τ D T)$ gives the mapping

$\begin{pmatrix} q\\ p \end{pmatrix} \mapsto \begin{pmatrix} q'\\ p' \end{pmatrix} = \begin{pmatrix} q + \tau c_i \frac{\partial T}{\partial p}(p)\\ p \end{pmatrix}$

and $exp(d i τ D V)$ gives

$\begin{pmatrix} q\\ p \end{pmatrix} \mapsto \begin{pmatrix} q'\\ p' \end{pmatrix} = \begin{pmatrix} q \\ p - \tau d_i \frac{\partial V}{\partial q}(q)\\ \end{pmatrix}.$

Note that both of these maps are practically computable.

The coefficients for the first order integrator of the SI scheme, with $k$ being 1, are trivially shown by

$c_1 = d_1 = 1. \ \$

The coefficients for the second order integrator, with $k$ being 2, are given by

$c_1 = c_2 = 1/2, \ \$

$d_1 = 1,\ \ d_2 = 0$

The fourth order integrator, with $k$ being 4, has the coefficients

$c_1 = c_4 = \frac{1}{2(2-2^{1/3})},\ \ c_2=c_3=\frac{1-2^{1/3}}{2(2-2^{1/3})},$

$d_1 = d_3 = \frac{1}{2-2^{1/3}},\ \ d_2 = -\frac{2^{1/3}}{2-2^{1/3}},\ \ d_4 = 0$

To determine these coefficients, the Baker-Campbell-Hausdorff (BCH) formula can be used. For the derivations of these values, and also for the determination of the coefficients of the higher order integrators, see Ref. [2].

[edit] References

H. Yoshida, Celestial Mechanics and Dynamical Astronomy 56, 27 (1993)
H. Yoshida, Phys. Lett. A 150, 262 (1990)
Ben Leimkuhler and Sebastian Reich, Simulating Hamiltonian Dynamics. Cambridge University Press, 2005. ISBN 0-521-77290-7.

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Categories: Numerical differential equations | Hamiltonian mechanics

Symplectic integrator

From Wikipedia, the free encyclopedia

[edit] Introduction

[edit] Splitting methods for separable Hamiltonians

[edit] References

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