- Symplectic integrator
In
mathematics , a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related toclassical mechanics andsymplectic geometry . Symplectic integrators form the subclass ofgeometric integrator s which, by definition, arecanonical transformation s. They are widely used inmolecular dynamics , discrete element methods, accelerator physics, and celestial mechanics.Introduction
Symplectic integrators are designed for the numerical solution of
Hamilton's equations , which read:dot p = -frac{partial H}{partial q} quadmbox{and}quad dot q = frac{partial H}{partial p},where q denotes the position coordinates, p the momentum coordinates, and H is the Hamiltonian (seeHamiltonian mechanics for more background).The time evolution of
Hamilton's equations is asymplectomorphism , meaning that it conserves the symplectictwo-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.
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). qquadqquad (1)This happens frequently in Hamiltonian mechanics, with "T" being the
kinetic energy and "V" thepotential 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 , and z=(q,p). 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( au)=exp( au D_H)z(0). (3)When the Hamiltonian has the form of eq. (1), the solution (3) is equivalent to:z( au) = exp [ au (D_T + D_V)] z(0). (4)The SI scheme approximates the time-evolution operator exp [ au (D_T + D_V)] in the formal solution (4) by a product of operators as:exp [ au (D_T + D_V)] = Pi_{i=1}^k exp(c_i au D_T)exp(d_i au D_V) + O( au^{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 au D_T) and exp(d_i au D_V) provides a symplecticmap, so their product appearing in the right hand side of (5) also constitutes a symplectic map. In concrete terms, exp(c_i au D_T) gives the mapping:egin{pmatrix}q\ pend{pmatrix}mapstoegin{pmatrix}q'\ p'end{pmatrix}=egin{pmatrix} q + au c_i frac{partial T}{partial p}(p)\ pend{pmatrix}and exp(d_i au D_V) gives:egin{pmatrix}q\ pend{pmatrix}mapstoegin{pmatrix}q'\ p'end{pmatrix}=egin{pmatrix} q \ p - au 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 = frac12, qquad d_1 = 1, qquad d_2 = 0.A fourth order integrator (with k=4) was independently discovered by three groups [cite journal
last = Forest
first = E.
coauthors=Ruth, R.D.
title=Fourth-order symplectic integration
journal=Physica D
date=1990
volume=43
pages=105
doi=10.1016/0167-2789(90)90019-L] [cite journal
last = Yoshida
first = H.
title = Construction of higher order symplectic integrators
journal=Phys. Lett. A
date=1990
volume=150
pages=262
doi = 10.1016/0375-9601(90)90092-3] [cite journal
last=Candy
first=J.
coauthors=Rozmus, W.
title=A Symplectic Integration Algorithm for Separable Hamiltonian Functions
journal=J. Comput. Phys.
date=1991
volume=92
pages=230
doi=10.1016/0021-9991(91)90299-Z]: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.References
* cite book
last = Leimkuhler
first = Ben
coauthors = Sebastian Reich
title = Simulating Hamiltonian Dynamics
publisher = Cambridge University Press
date = 2005
id = ISBN 0-521-77290-7
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