NationMaster - Encyclopedia: Symplectic integrator

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, accelerator physics, and celestial mechanics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In the mathematical field of numerical ODEs, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation. ... Molecular dynamics (MD) is a form of computer simulation wherein atoms and molecules are allowed to interact for a period of time under known laws of physics, giving a view of the motion of the atoms. ... The term discrete element method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains of sand. ...

Introduction

Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ...

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

where $q$ denotes the position coordinates, $p$ the momentum coordinates, and $H$ is the Hamiltonian (see Hamiltonian mechanics for more background). Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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. In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

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. Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... Molecular dynamics (MD) is a form of computer simulation wherein atoms and molecules are allowed to interact for a period of time under known laws of physics, giving a view of the motion of the atoms. ...

Most of the usual numerical methods, like the primitive Euler scheme and the classical Runge-Kutta scheme, are not symplectic integrators. In mathematics and computational science, Euler integration is the most basic kind of numerical integration for calculating trajectories from forces at discrete timesteps. ... In numerical analysis, the Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. ...

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 the potential energy. The kinetic energy of an object is the extra energy which it possesses due to its motion. ... {{Portal|Energy}Potential energy is the energy available within a physical system due to an objects position in conjunction with a conservative force which acts upon it (such as the gravitational force or Coulomb force). ...

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 symplectic Euler method is the first-order integrator with $k = 1$ and coefficients In mathematics, the Eulerâ€“Cromer algorithm or symplectic Euler method is a modification of the Euler method for solving Hamiltons equations, a system of ordinary differential equations that arises in classical mechanics. ...

$c_1 = d_1 = 1.$

The Verlet method is the second-order integrator with $k = 2$ and coefficients Verlet integration is a method for calculating the trajectories of particles in molecular dynamics simulations. ...

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

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. In mathematics, the Baker-Campbell-Hausdorff formula is the solution to for non-commuting x and y. ...

References

^ Forest, E.; Ruth, R.D. (1990). "Fourth-order symplectic integration". Physica D 43: 105.
^ Yoshida, H. (1990). "Construction of higher order symplectic integrators". Phys. Lett. A 150: 262.
^ 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.

Categories: Numerical differential equations | Hamiltonian mechanics

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