In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.
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. (See Hamiltonian mechanics for more background.)
The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2form . A numerical scheme is a symplectic integrator if it also conserves this 2form.
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 longterm evolution of chaotic Hamiltonian systems ranging from the Kepler problem to the classical and semiclassical 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.
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

(1)
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 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

(2)
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 a matrix exponential:

(3)
Note the positivity of in the matrix exponential.
When the Hamiltonian has the form of equation (1), the solution (3) is equivalent to

(4)
The SI scheme approximates the timeevolution operator in the formal solution (4) by a product of operators as

(5)
where and are real numbers, is an integer, which is called the order of the integrator, and where . Note that each of the operators and provides a symplectic map, so their product appearing in the righthand side of (5) also constitutes a symplectic map.
Since for all , we can conclude that

(6)
By using a Taylor series, can be expressed as

(7)
where is an arbitrary real number. Combining (6) and (7), and by using the same reasoning for as we have used for , we get

(8)
In concrete terms, gives the mapping
and gives
Note that both of these maps are practically computable.
Examples
The simplified form of the equations (in executed order) are:
After converting into Lagrangian coordinates:
Where is the force vector at , is the acceleration vector at , and is the scalar quantity of mass.
Several symplectic integrators are given below. An illustrative way to use them is to consider a particle with position and velocity .
To apply a timestep with values to the particle, carry out the following steps:
Iteratively:
 Update the position of the particle by adding to it its (previously updated) velocity multiplied by
 Update the velocity of the particle by adding to it its acceleration (at updated position) multiplied by
A firstorder example
The symplectic Euler method is the firstorder integrator with and coefficients
Note that the algorithm above does not work if timereversibility is needed. The algorithm has to be implemented in two parts, one for positive time steps, one for negative time steps.
A secondorder example
The Verlet method is the secondorder integrator with and coefficients
Since , the algorithm above is symmetric in time. There are 3 steps to the algorithm, and step 1 and 3 are exactly the same, so the positive time version can be used for negative time.
A thirdorder example
A thirdorder symplectic integrator (with ) was discovered by Ronald Ruth in 1983.^{[1]} One of the many solutions is given by
A fourthorder example
A fourthorder integrator (with ) was also discovered by Ruth in 1983 and distributed privately to the particleaccelerator community at that time. This was described in a lively review article by Forest.^{[2]} This fourthorder integrator was published in 1990 by Forest and Ruth and also independently discovered by two other groups around that same time.^{[3]}^{[4]}^{[5]}
To determine these coefficients, the Baker–Campbell–Hausdorff formula can be used. Yoshida, in particular, gives an elegant derivation of coefficients for higherorder integrators. Later on, Blanes and Moan^{[6]} further developed partitioned Runge–Kutta methods for the integration of systems with separable Hamiltonians with very small error constants.
Splitting methods for general nonseparable Hamiltonians
General nonseparable Hamiltonians can also be explicitly and symplectically integrated.
To do so, Tao introduced a restraint that binds two copies of phase space together to enable an explicit splitting of such systems.^{[7]} The idea is, instead of , one simulates , whose solution agrees with that of in the sense that .
The new Hamiltonian is advantageous for explicit symplectic integration, because it can be split into the sum of three subHamiltonians, , , and . Exact solutions of all three subHamiltonians can be explicitly obtained: both solutions correspond to shifts of mismatched position and momentum, and corresponds to a linear transformation. To symplectically simulate the system, one simply composes these solution maps.
Applications
In plasma physics
In recent decades symplectic integrator in plasma physics has become an active research topic,^{[8]} because straightforward applications of the standard symplectic methods do not suit the need of largescale plasma simulations enabled by the peta to exascale computing hardware. Special symplectic algorithms need to be customarily designed, tapping into the special structures of physics problem under investigation. One such example is the charged particle dynamics in an electromagnetic field. With the canonical symplectic structure, the Hamiltonian of the dynamics is
A more elegant and versatile alternative is to look at the following noncanonical symplectic structure of the problem,
See also
References
 ^ Ruth, Ronald D. (August 1983). "A Canonical Integration Technique". IEEE Transactions on Nuclear Science. NS30 (4): 2669–2671. Bibcode:1983ITNS...30.2669R. doi:10.1109/TNS.1983.4332919.
 ^ Forest, Etienne (2006). "Geometric Integration for Particle Accelerators". J. Phys. A: Math. Gen. 39 (19): 5321–5377. Bibcode:2006JPhA...39.5321F. doi:10.1088/03054470/39/19/S03.
 ^ Forest, E.; Ruth, Ronald D. (1990). "Fourthorder symplectic integration" (PDF). Physica D. 43: 105–117. Bibcode:1990PhyD...43..105F. doi:10.1016/01672789(90)90019L.
 ^ Yoshida, H. (1990). "Construction of higher order symplectic integrators". Phys. Lett. A. 150 (5–7): 262–268. Bibcode:1990PhLA..150..262Y. doi:10.1016/03759601(90)900923.
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 ^ Qin, H.; Guan,X. (2008). "A Variational Symplectic Integrator for the Guiding Center Motion of Charged Particles for Long Time Simulations in General Magnetic Fields" (PDF). Physical Review Letters. 100: 035006. doi:10.1103/PhysRevLett.100.035006. PMID 18232993.
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