Constrained Dynamics

Spring A Bead on a Wire x f

Spring A Bead on a Wire x f Zero rate-of- change ensures constant C!

Spring A Bead on a Wire x f

Spring A Bead on a Wire x f

Spring A Bead on a Wire x f However, one equation cannot solve two unknowns of f- hat!?

Spring A Bead on a Wire x f Constraint force does not produce work

Spring Constrained Dynamics General Recipe

Spring Constrained Dynamics

Spring About Constraint Force … Q q Constraint force Q-hat induced by Q to move q along legal velocity q-dot

Spring About Constraint Force … Q q Legal velocity q-dot is  to the hyperplane spanned by the gradients of all constraints. Hence, constraint force is a linear combination of all gradients.

Spring Numerical Concerns To account for numerical drift, we often apply numerical damping as follows:

Spring Ex: Bead in Recipe x Q

Spring Example: Double Pendulum l1l1 l2l2 p1p1 p2p2