This video is an introduction to trajectory optimization, with a special focus on direct collocation methods. The slides are from a presentation that I gave at Cornell, linked here:
http://www.matthewpeterkelly.com/tutorials/trajectoryOptimization/cartPoleCollocation.svg
The journal paper version of this talk, to be published by SIAM Review in December 2017:
http://www.matthewpeterkelly.com/research/MatthewKelly_IntroTrajectoryOptimization_SIAM_Review_2017.pdf
The derivation of the cart-pol dynamics can be found here:
http://www.matthewpeterkelly.com/tutorials/cartPole/
My general tutorial page for trajectory optimization is:
http://www.matthewpeterkelly.com/tutorials/trajectoryOptimization/