I understand this question has been asked many times and I have thoroughly looked at the previous questions so please read my question carefully before flagging it as a duplicate.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically this means that, if all the co-ordinates $q$ and $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined.
Now I am a bit confused, is this saying that we only need to know the initial position and initial velocity to get the full time evolution of the system? which is due to the fact that the dynamics are governed by a second order differential equation?
I just want to give an example here and please tell me if I am correct in my thought process:
Take the example of a spring that obeys Hooke's law: $$F = m \ddot{x}= -kx$$ then a general solution can be: $$x(t) = A e^{i \sqrt{\frac{k}{m}}t}+ B e^{-i \sqrt{\frac{k}{m}}t}$$
so we do not need velocities or position for all time, for a given particle we just need initial conditions for velocity and position and then we are set?
my last question is why can we rearrange Newton's law in the form $$\ddot{x} = F(x,\dot{x},t)/m$$ How do we know for a fact that $F$ can generally depend $x,\dot{x},t$ and that is it?
Possible duplicate list:
Why are position and velocity enough for prediction and acceleration is unnecessary?
Why forces are functions of time, position and velocity at most?
Why are there only derivatives to the first order in the Lagrangian?
Why are differential equations for fields in physics of order two?
