Example 4: Pendulum Oscillations

The position ( at time of a particle of mass oscillating on a pendulum of length is

where is the acceleration due to gravity, is the tension in the string and is the angle of the pendulum relative to the vertical at time . These equations are however insufficient to guarantee that the particle stays on the string. To ensure that this is so, we must supplement the ODEs by the algebraic constraint.

The two second order differential equations and the constraint comprise a system of three differential algebraic equations (DAE's) for the unknowns .

Initial conditions specify , and , but not .

Note: Of course, for this problem, it is easy to eliminate the constraint by introducing the change of variables

This would reduce the DAEs to the second order ODE

IVP's will comprise our initial study and followed by a brief discussion of BVP's.