How can the whole path of a particle be determined by its configuration at any time and the rate of change of configuration at that time? 
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?
 A: I am assuming you are referring to the part that talks about Newton's laws of motion are second order in time. This is talking about the differential equation
$$\mathbf F=m\mathbf {\ddot x}$$
where each dot above $\mathbf x$ is shorthand for a time derivative. The equation is second order because the highest order derivative is a second derivative. It is a well known property of second order ordinary differential equations that specifying $\mathbf x(t_1)$ and $\mathbf {\dot x}(t_1)$ is sufficient to determine a unique solution to this differential equation.
Now, in introductory physics classes you just deal with position $\mathbf x$, but when you start dealing with Lagrangian mechanics you can use generalized coordinates $$\mathbf q(t)=(q_1(t), q_2(t),\dots ,q_n(t))$$
But it can be shown that this method is equivalent to Newton's laws, so the idea of a second order differential equation still holds.
These generalized coordinates are the coordinates of our configuration space. So when we talk about "paths" in this context we don't necessarily have to be talking about actual paths in physical space. We are talking about a path in this configuration space defined by the generalized coordinates 
Also, all derivatives here are showing respect to time.
