In the image, it says the whole path can be determined by knowing u(t1) and u'(t1) at any point t1. As far as I know, using u(t1) and u'(t1), the best we can do is approximate a nearby point u(t1+del t)= u(t1)+ u'(t1)*del t. Don't we need all the derivatives to calculate the whole path using Taylor polynomial?
Also, what kind of derivative is u'(t) here? The way I understand it, u(t) is the configuration function of the body. The output of u(t) is six numbers (center of mass co-ordinates, and three euler angles). How do we even take the derivative of such a function? Isn't derivative 'change in output' divided by infinitesimal change in input. How is change in output defined here?