[Physics] Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system

classical-mechanicscoordinate systemslagrangian-formalismvelocity

I am a Physics undergraduate, so provide references with your responses.

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.

They justify this as being "known from experience", which is not entirely satisfactory. What is the basis for their assertion?

Similar: Why are there only derivatives to the first order in the Lagrangian?

Is his question equivalent to mine, even though his solely refers to Lagrangian Mechanics?

Furthermore, this might just indicate how mathematically crude my mind is, but why is it not sufficient simply to give the coordinates $q$, and determine $\dot{q}$ from that, i.e. if $q$ is given by some smooth function, can we not determine all further derivatives from that alone?

Best Answer

You should think of this by timestepping Newton's laws--- if you know the positions and velocity and one instant, you know the force, and the force determines the acceleration. This allows you to determine the velocity and an infinitesimal time in the future by

$$ v(t+dt) = v(t) + dt F/m $$ $$ x(t+dt) = x(t) + dt v $$

You then find the position and velocity at the next time step, and you find the new force, and continue forever. This is an algorithm to solve Newton's laws, and all that LL are saying is that Newton's laws are known from experience with objects, they are inducted from observations.

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