Differential-algebraic system of equations

Initial conditions must be a solution of the system of equations of the form

restricting the direction of motion to the tangent of the circle. The next derivative of this equation implies

To obtain unique derivative values for all dependent variables the last equation was three times differentiated. This gives a differentiation index of 3, which is typical for constrained mechanical systems.

Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.

Once the model has been converted to algebraic equation form, it is solvable by large-scale nonlinear programming solvers (see APMonitor).