402 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
sum is defined by
N-l
§ = L lL (vk+1, vk).
k=OExtremize § : v N+l -t JR subject to the constraint that Vk E Q c v for k
1, · · · , N - 1, i.e., solveD21L (vk+I, vk) + D11L (vk, vk-1) + >..f Dg (vk) = 0
(no sum on k) with g (vk) = 0 for k = 1, · · · , N - l. Here >..k are Lagrange multi-
pliers, chosen to enforce the constraints.Summary. The algorithm is defined by starting with Vk and Vk-l in QC V, i.e.,
g (vk) = 0 and g (vk-l) = 0, and solvingD21L (vk+1, vk) + D11L (vk, vk-1) + >..k T Dg (vk) = 0
subject tog (vk+l) = 0, for vk+ 1 and >..k.
In terms of the unconstrained Lagrangian, the algorithm reads as follows:2 [aL (Vk + Vk-1 Vk - Vk-1) aL (Vk+l + Vk Vk+l - Vk )]
h av 2 ' h av 2 ' h
~ [aL (Vk + Vk-1 Vk - Vk-1) a L (Vk+l + Vk Vk+l - Vk )]+ 2 av 2 ' h + av 2 ' h
+DgT (vk)>..k=O
together with g (vk+ 1 ) = 0.Example. If the continuous Lagrangian is
L(q,q) = ~qTMq-V(q)
wit h constraint g(q) = 0, where Mis a constant mass matrix, and Vis the potential
energy, t hen the DEL equations areM (Vk+l -2vk +vk-1) ~(av (Vk+l +vk) av (Vk +vk-1))
h^2 + 2 aq 2 + aq 2
-D7g (vk)>..k = 0
wit h g (vk+ 1 ) = 0.
These algorithms produce the Verlet and sh ake algorithms as sp ecial cases, as is
disc ussed in Wendlandt and Marsden [1997]. The following result is very plausible:
it states that the earlier intrinsic construction is coincident with this construction
using constraints.Theorem 5.4. The algorithm defined using Lagrange multipliers coincides with
that defined intrinsically using the constrained discrete Lagrangian on Q x Q, so it
is symplectic and momentum preserving.The examples treated in Wendlandt and Marsden [1997] are simple ones, the
double spherical pendulum ( dsp) and the rigid body. The former example is inter-
esting b ecause of its chaotic dynamics and pattern evocation phenomenon (Marsden
and Scheurle [199 5 ]).
Both are constrained systems and may be readily integrated by the above meth-
ods. Numerical issues such as cpu time and accuracy a re also discussed, as well as