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 geometric integrators by considering the motion of a
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
.
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 may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
of
the system, the sum of its
kinetic and
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
energies, is
:
which gives
Hamilton's equations
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 ...
:
It is natural to take the
configuration space 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 arbitrary non-negative integer
we write
.
We use the following methods.
:
(
explicit Euler),
:
(
implicit Euler),
:
(
symplectic Euler),
:
(
implicit midpoint rule
Implicit may refer to:
Mathematics
* Implicit function
* Implicit function theorem
* Implicit curve
* Implicit surface
* Implicit differential equation
Other uses
* Implicit assumption, in logic
* Implicit-association test, in social psychology ...
).
(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 of
. We plot, in
, the exact
trajectories and the numerical solutions of the system. For the explicit
and implicit Euler methods we take
, and ''z''
0 = (0.5, 0) and (1.5, 0) respectively; for the other two methods we take
, and ''z''
0 = (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) integrators.
Moving frame method
The
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
In lay te ...
method can be used to construct numerical methods which preserve
Lie symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of the ODE. Existing methods such as
Runge-Kutta can be modified using moving frame method to produce invariant versions.
[ Pilwon Kim (2006), ]
Invariantization of Numerical Schemes Using Moving Frames
See also
*
Energy drift
*
Mimesis (mathematics)
References
Further reading
*
*
*
*
{{DEFAULTSORT:Geometric Integrator
Numerical differential equations