In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a symplectic integrator (SI) is a
numerical integration scheme for
Hamiltonian systems. Symplectic integrators form the subclass of
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
We can motivate the study of g ...
s which, by definition, are
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...
s. They are widely used in
nonlinear dynamics
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
,
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
,
discrete element methods,
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
,
,
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, and
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
.
Introduction
Symplectic integrators are designed for the numerical solution of
Hamilton's equations, which read
:
where
denotes the position coordinates,
the momentum coordinates, and
is the Hamiltonian.
The set of position and momentum coordinates
are called
canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
.
(See
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
for more background.)
The time evolution of
Hamilton's equations is a
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
, meaning that it conserves the symplectic
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
. A numerical scheme is a symplectic integrator if it also conserves this 2-form.
Symplectic integrators also might possess, as a conserved quantity, a Hamiltonian which is slightly
perturbed from the original one (only true for a small class of simple cases). 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
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
to the classical and semi-classical simulations in
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
.
Most of the usual numerical methods, like the primitive
Euler scheme and the classical
Runge–Kutta scheme, are not symplectic integrators.
Methods for constructing symplectic algorithms
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
In physics, the kinetic energy of an object is the energy that 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 acce ...
and ''V'' the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
.
For the notational simplicity, let us introduce the symbol
to denote the canonical coordinates
including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be expressed in a single expression as
where
is a
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. T ...
. Furthermore, by introducing an operator
, which returns a
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. T ...
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 a
matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
:
Note the positivity of
in the matrix exponential.
When the Hamiltonian has the form of equation (), the solution () is equivalent to
The SI scheme approximates the time-evolution operator