LECTURE 5. VARIATIONAL INTEGRATORS 403

the comparison of these methods with energy-momentum methods. For the rigid

body, we use quaternions to handle the constraints but this was just for illustration

and of course one co uld handle the constraints directly using 80(3) c GL(3) as

well. The following figure from one of the dsp calculations shows the surrogate

energy phenomenon.

I • I

s 10 ts 20 2S 30

h = O.ls time (s) h = O.Ols

h =0.00ls time (s) h = O.OOOls

=~ 0

o s 10 ts 20 2S 30

time (s)

Figure 5.1. Energy and multipliers versus time for the dsp simulation.

An intrinsic variational viewpoint. Recall that given a Lagrangian function
L : TQ -+ JR, we construct the corresponding action functional 6 on C^2 curves
q(t) by (using coordinate notation)

rb ( d i )

6(q(-)) =la L qi (t), d~ (t) dt. (5.17)

The action functional depends on a and b, but this is not explicit in the notation.
Hamilton's principle seeks the curves q(t) for which the functional 6 is station-

ary under variations of qi (t) with fixed endpoints. It will be useful to recall this

calculation ; namely, we seek curves q(t) which satisfy

d6(q(t)) · 8q(t) := :E lc=O 6(q(t) + EOq(t)) = 0 (5.18)