−1 −0.5 0 0.5
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
z coordinate
z momentum
Figure 2-10: A contour of the reduced Hamiltonian for the spherical pendulum.
should try to minimize the error in a number of conservation laws. This is what is done with the example of rigid body motion.
Finally, Figure 2-10 shows more evidence that this integrator has found the correct solution: Since angular momentum is conserved for the spherical pendulum, we know that the angular motion (about the vertical axis) of the pendulum may be decoupled from its vertical motion, and the system may be reduced to one with a lower degree of freedom. In this gure, the
z
vs.p
zplot shows that the trajectory of the reduced system is a closed curve.This is because energy is also conserved in the reduced system, and hence trajectories of the reduced system must follow equipotential curves of the reduced Hamiltonian.
ψ
θ
ϕ z
x
Figure 2-11: Euler angles for a rigid body.
Furthermore, the conguration space for the rotational motion of rigid bodies is the space of all orientations of a rigid body, or equivalently the space of all rotation matrices in three dimensions.17 The manifold structure of this space is rather abstract, and since it is really a 3-manifold imbedded in the 9-dimensional space of all 33 matrices, we can no longer rely on our geometric intuition to approach this problem. This is one of the most important examples of an abstract manifold.
Traditionally, orientations of rigid bodies are described by Euler angles, depicted in Fig-ure 2-11. As hinted at earlier, this coordinate system has the problem that the coordinates
\blow up" (the Jacobian of the coordinate map becomes singular) when the rigid body is standing vertically, as a bit of analysis will show. This is known as a coordinate singularity because the singularity is part of the coordinate system, not a feature of the dynamics.
The traditional approach to this problem is to work entirely in Euler angles. This works well so long as the trajectory does not come near the coordinate singularity. But when it does, the singularity can have a serious eect on numerical accuracy, which is often reected in uctuations in the conserved quantities. In this example, the results of a numerical integration of rigid body motion is presented using the traditional and the
17This space is commonly denoted asSO3, thespecial orthogonal group. It is an example of aLie group, which are manifolds that also happen to be groups, and where the group operations are smooth as maps on manifolds.
0 2000 4000 6000 8000 10000
−8
−6
−4
−2 0 2 4 6 8 10x 10−14
time index
relative error
Figure 2-12: Relative error in energy conservation for rigid body motion in Euler angles.
manifold method. The principal moments of inertia of the rigid body are 1, p2, and 2, with mass set to
m
= 1. The initial conditions, in Euler angles, are = 0, = 1, = 0, _ = ;0:
01, _ = ;0:
1, and _ = ;0:
01; these initial conditions have been chosen to take the trajectory close to the coordinate singularity in Euler angles, so that the eects of the singularity on conserved quantities can be observed. The integration was performed using a time step of 0.01, for 100.0 time units (which equals 10,000 time steps). The integration in Euler angles used a Bulirsch-Stoer integrator, which the manifold integrator also used as its local integrator.Figure 2-12 shows the relative error in energy conservation for a trajectory that comes relatively near the singularity. Figure 2-13 shows the analogous plot for the manifold method.
In Figure 2-12, the maximum absolute value is 8
:
43194301271212 10;14, and the corresponding average is 2:
6428202894715013 10;14. In contrast, in Figure 2-13, the maximum absolute value of the error is 1:
39438746319169310;14, and the average absolute value of the error is 4:
31070783106112 10;15. Thus, the manifold approach actually conserves energy better: In terms of relative error, it outperforms the traditional approach0 2000 4000 6000 8000 10000
−1.5
−1
−0.5 0 0.5
1x 10−14
time index
relative error
Figure 2-13: Relative error in energy conservation for rigid body motion using the manifold approach.
0 2000 4000 6000 8000 10000
−5
−4
−3
−2
−1 0 1 2 3 4x 10−14
time index
relative error
Figure 2-14: Relative error in conserving the
x
component of the angular momen-tum for rigid body motion using Euler angles. The maximum absolute value of the error is 4:
163336342344337 10;14, while the average absolute value of the error is 1:
997547960375101210;14.by about six times.
Note that in Figure 2-12, the curve has a rather sharp peak at time index 4000. That is a consequence of a close encounter between the trajectory and the coordinate singularity.
Such a peak can be seen in all of the following plots that were generated using the Euler angles (Figures 2-14, 2-16, and 2-18), and are absent from the plots generated by using the manifold integrator (Figures 2-13, 2-15, 2-17, and 2-19).
Similar comparisons can be made using the components of the angular momentum, as shown in Figures 2-14 through 2-19.
In contrast to the spherical pendulum, in this example all the components of angular momentum (as computed from the inertial frame), as well as the energy function, are used in the integration. Thus, the manifold integrator attempts to minimize deviations from initial values of conserved quantities, which improves their conservation at the cost of making it harder to check how well the system does.
0 2000 4000 6000 8000 10000
−3
−2
−1 0 1 2 3 4x 10−15
time index
relative error
Figure 2-15: Relative error in conserving the
x
component of the angular momentum for rigid body motion using the manifold approach. The maximum absolute value of the error is 2:
7755575615628914 10;15, while the average absolute value of the error is 3:
874817133819874410;16.0 2000 4000 6000 8000 10000
−2
−1 0 1 2 3 4 5x 10−13
time index
relative error
Figure 2-16: Relative error in conserving the
y
component of the angular momentum for rigid body motion using Euler angles. The maximum absolute value of the er-ror is 4:
3375450673823944 10;13, while the average absolute value of the error is 8:
34873481018122910;14.0 2000 4000 6000 8000 10000
−2
−1 0 1 2x 10−14
time index
relative error
Figure 2-17: Relative error in conserving the
y
component of the angular momentum for rigid body motion using the manifold approach. The maximum absolute value of the error is 1:
7798707703661857 10;14, while the average absolute value of the error is 2:
108345921037557610;15.0 2000 4000 6000 8000 10000
−3
−2
−1 0 1 2 3 4 5x 10−13
time index
relative error
Figure 2-18: Relative error in conserving the
z
component of the angular momen-tum for rigid body motion using Euler angles. The maximum absolute value of the error is 4:
352060412667006 10;13, while the average absolute value of the error is 9:
47999272470440410;14.0 2000 4000 6000 8000 10000
−1.5
−1
−0.5 0 0.5 1 1.5x 10−14
time index
relative error
Figure 2-19: Relative error in conserving the