Symplectic integrator
Encyclopedia
In mathematics
, a symplectic integrator (SI) is a numerical integration scheme
for a specific group of differential equation
s related to classical mechanics
and symplectic geometry. Symplectic integrators form the subclass of geometric integrator
s which, by definition, are canonical transformation
s. They are widely used in molecular dynamics
, discrete element method
s, accelerator physics
, and celestial mechanics
.
where
denotes the position coordinates, 
the momentum coordinates, and 
is the Hamiltonian.
The set of position and momentum coordinates
are called canonical coordinates
.
(See Hamiltonian mechanics
for more background.)
The time evolution of Hamilton's equations is a symplectomorphism
, meaning that it conserves the symplectic two-form
. 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.
Assume that the Hamiltonian is separable, meaning that it can be written in the form
This happens frequently in Hamiltonian mechanics, with T being the kinetic energy
and V the potential energy
.
For the notational simplicity, let us introduce the symbol
to denote the canonical coordinates
including both of the position and momentum coordinates.
Then, the set of the Hamiltonian's equations given in the introduction can be expressed in a single expression as
where
is a Poisson bracket
.
Furthermore, by introducing an operator,
, which returns
a Poisson bracket
of the operand with the Hamiltonian
, the expression of the Hamilton's equation
can be further simplified to
The formal solution of this set of equations is given as
When the Hamiltonian has the form of eq. (1), the solution (3) is equivalent to
The SI scheme approximates the time-evolution operator
in the formal solution (4) by a product of operators as
where
and 
are real numbers, and 
is an
integer, which is called the order of the integrator. Note that each of the operators
and 
provides a symplectic
map, so their product appearing in the right hand side of (5) also constitutes a
symplectic map. In concrete terms,
gives the mapping
and
gives
Note that both of these maps are practically computable.
The symplectic Euler method is the first-order integrator with
and coefficients
The Verlet method
is the second-order integrator with
and coefficients
A third order symplectic integrator (with
) was discovered by Ronald Ruth in 1983.
One of the many solutions is given by
A fourth order integrator (with
) was also discovered by Ruth in 1983 and distributed privately to the
accelerator community at that time. This was described in a lively review article by Forest.
This fourth order integrator was published in 1990 by Forest and Ruth and also
independently discovered by two other groups around that same time
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a symplectic integrator (SI) is a numerical integration scheme
Numerical ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
for a specific group of differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s related to classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
and symplectic geometry. Symplectic integrators form the subclass of geometric integrator
Geometric integrator
In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.-Pendulum example:...
s which, by definition, are canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates → that preserves the form of Hamilton's equations , although it...
s. They are widely used in molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
, discrete element method
Discrete element method
A discrete element method , also called a distinct element method is any of family of numerical methods for computing the motion of a large number of particles of micrometre-scale size and above...
s, accelerator physics
Particle accelerator
A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
, and celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
.
Introduction
Symplectic integrators are designed for the numerical solution of Hamilton's equations, which readwhere
denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.The set of position and momentum coordinates
are called canonical coordinatesCanonical coordinates
In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics...
.
(See Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
for more background.)
The time evolution of Hamilton's equations is a symplectomorphism
Symplectomorphism
In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...
, meaning that it conserves the symplectic two-form
Two-form
In linear algebra, a two-form is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skew-symmetric....
. 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
Perturbation theory
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...
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
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
.
Most of the usual numerical methods, like the primitive Euler scheme
Euler integration
In mathematics and computational science, the Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations with a given initial value...
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
This happens frequently in Hamiltonian mechanics, with T being the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
and V the potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
.
For the notational simplicity, let us introduce the symbol
to denote the canonical coordinatesincluding both of the position and momentum coordinates.
Then, the set of the Hamiltonian's equations given in the introduction can be expressed in a single expression as
where
is a Poisson bracketPoisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
.
Furthermore, by introducing an operator,
, which returnsa Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
of the operand with the Hamiltonian
Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
, the expression of the Hamilton's equation
can be further simplified to
The formal solution of this set of equations is given as
When the Hamiltonian has the form of eq. (1), the solution (3) is equivalent to
The SI scheme approximates the time-evolution operator
in the formal solution (4) by a product of operators aswhere
and are real numbers, and is aninteger, which is called the order of the integrator. Note that each of the operators
and provides a symplecticmap, so their product appearing in the right hand side of (5) also constitutes a
symplectic map. In concrete terms,
gives the mappingand
givesNote that both of these maps are practically computable.
The symplectic Euler method is the first-order integrator with
and coefficientsThe Verlet method
Verlet integration
Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics...
is the second-order integrator with
and coefficientsA third order symplectic integrator (with
) was discovered by Ronald Ruth in 1983.One of the many solutions is given by
A fourth order integrator (with
) was also discovered by Ruth in 1983 and distributed privately to theaccelerator community at that time. This was described in a lively review article by Forest.
This fourth order integrator was published in 1990 by Forest and Ruth and also
independently discovered by two other groups around that same time
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.