Baumgarte's Method for Holonomic Constraints

Let us consider first a dynamical system which is subject to m holonomic constraints of the form

(6.284)

Suppose there are n second-order dynamical equations written in the fundamental La-grangian form

(6.285)

where the ?s are Lagrange multipliers, the Qs are generalized applied forces, and where

(6.286)

At this point there are n dynamical equations which are linear in the n s and the m ?s. We need m additional equations to solve for the variables. These additional equations can be obtained by differentiating the constraint equations twice with respect to time. First,

(6.287)

where

(6.288)

Then has the form

(6.289)

If these expressions for are set equal to zero then, with the aid of (6.285), one can solve for the ( q, , t) and ? _j( q, , t). The n expressions can be integrated numerically for given initial conditions, thereby obtaining and q _i as functions of time.

The problem with this approach is that the resulting numerical solutions will be unstable even though the physical system may actually...