Symplectic integrator

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In mathematics, a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are cannonical transformations. They are widely used in molecular dynamics, discrete element methods, accelerator physics, 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(DT + DV)] 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 ci and di are real numbers, and n is an integer, which is called the order of the integrator. Note that each of the operators exp(ciτDT) and exp(diτDV) provides a symplectic map, so their product appearing in the right hand side of (5) also constitutes a symplectic map. In concrete terms, exp(ciτDT) 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(diτDV) 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 symplectic Euler method is the first-order integrator with k = 1 and coefficients

c_1 = d_1 = 1. \ \

The Verlet method is the second-order integrator with k = 2 and coefficients

c_1 = c_2 = \tfrac12, \qquad d_1 = 1, \qquad d_2 = 0.

A fourth order integrator (with k = 4) was independently discovered by three groups [1] [2] [3]

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 formula can be used. Yoshida, in particular, gives an elegant derivation of coefficients for higher-order integrators.

[edit] References

  1. ^ Forest, E.; Ruth, R.D. (1990). "Fourth-order symplectic integration". Physica D 43: 105. 
  2. ^ Yoshida, H. (1990). "Construction of higher order symplectic integrators". Phys. Lett. A 150: 262. 
  3. ^ Candy, J.; Rozmus, W. (1991). "A Symplectic Integration Algorithm for Separable Hamiltonian Functions". J. Comput. Phys. 92: 230. 
  • Leimkuhler, Ben; Sebastian Reich (2005). Simulating Hamiltonian Dynamics. Cambridge University Press. ISBN 0-521-77290-7.