Euler Poincare Dynamics and Control on Lie Groupoids

We study Euler Poincare dynamics on Lie groupoids with a focus on optimal control. Extending the classical Lie group formulation, we derive reduced equations on trivial Lie groupoids and interpret the result as a generalized rigid body on the sphere. This model couples internal rotational dynamics with translational motion on the Sphere. We illustrate the framework with an example from collective cell migration, showing how groupoid-based dynamics capture the trade-off between energy and migration time. The results demonstrate how Lie groupoid methods broaden the applicability of geometric control beyond standard rigid body systems