In the mathematical field of numerical ordinary differential equations

Numerical ordinary differential equations

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of 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 geometric integrators by considering the motion of a pendulum.

Assume that we have a pendulum whose bob has mass and
whose rod is massless of length . Take the
acceleration due to gravity to be . Denote by
the angular displacement of the rod from the vertical,
and by the pendulum's momentum. The Hamiltonian

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...

 of
the system, the sum of its kinetic

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 potential

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...

 energies, is


which gives Hamilton's equations


It is natural to take the configuration space

Configuration space

- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...

  of all to be the unit
circle , so that lies on the
cylinder . However, we will take
, simply because -space is
then easier to plot. Define
and . Let us experiment by
using some simple numerical methods to integrate this system. As usual,
we select a constant step size, , and for an aribtrary non-negative integer we write
.
We use the following methods.

(explicit Euler),

(implicit Euler),

(symplectic Euler),

(implicit midpoint rule).

(Note that the symplectic Euler method treats q by the explicit and by the implicit Euler method.)

The observation that is constant along the solution
curves of the Hamilton's equations allows us to describe the exact
trajectories of the system: they are the level curves

Level set

In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....

 of . We plot, in , the exact
trajectories and the numerical solutions of the system. For the explicit
and implicit Euler methods we take , and z₀ = (0.5, 0) and (1.5, 0) respectively; for the other two methods we take , and z₀ = (0, 0.7), (0, 1.4) and (0, 2.1).
The explicit (resp. implicit) Euler method spirals out from (resp. in to) the origin. The other two methods show the correct qualitative behaviour, with the implicit midpoint rule agreeing with the exact solution to a greater degree than the symplectic Euler method.

Recall that the exact flow of a Hamiltonian system with one degree of freedom is
area-preserving, in the sense that for all .
This formula is easily verified by hand. For our pendulum
example we see that the numerical flow of the explicit Euler method is not area-preserving; viz.,


A similar calculation can be carried out for the implicit Euler method,
where the determinant is


However, the symplectic Euler method is area-preserving:


thus . The implicit midpoint rule has similar geometric properties.

To summarize: the pendulum example shows that, besides the explicit and
implicit Euler methods not being good choices of method to solve the
problem, the symplectic Euler method and implicit midpoint rule agree
well with the exact flow of the system, with the midpoint rule agreeing
more closely. Furthermore, these latter two methods are area-preserving,
just as the exact flow is; they are two examples of geometric (in fact, symplectic

Symplectic integrator

In mathematics, a symplectic integrator is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations...

) integrators.

Moving frame method

The moving frame

Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...

 method can be used to construct numerical methods which preserve Lie

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 symmetries

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

 of the ODE. Existing methods such as Runge-Kutta can be modified using moving frame method to produce invariant versions.