Position and speed are enough in a physical system?

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I've recently read in this article http://www.en.uni-muenchen.de/news/newsarchiv/2013/f-m-77-13.html that "In the world of classical mechanics, the state of a physical system and its future evolution is fully determined by the instantaneous locations and velocities of its constituent particles"

Question 1.
Can someone provide me a pratical exemple that shows how we can derive everything from a physical system given the position and velocity of every particle?

I've got into this topic recently when i heard about "jerk", the derivative of acceleration, and wondered why we usually define equations using the second derivative, an exemple F=ma.

Question 2.
Why didnt Newton use jerk in his equations? Is jerk useless? Or is it that if we ignore jerk we can still have a good approximation of reality? I really need some answers here :(

I've also read here http://physics.stackexchange.com/questions/4102/why-are-there-only-derivatives-to-the-first-order-in-the-lagrangian the following
"To put it in simple terms, since Newton's second law relates functions which are two orders of derivative apart, you only need the 0th and 1st derivatives, position and velocity, to "bootstrap" the process, after which you can compute any higher derivative you want, and from that any physical quantity. This is analogous to (and in fact closely related to) the fact that to solve a second-order differential equation, you only need two initial conditions, one for the value of the function and one for its derivative."

Can someone better explain this to me? I can't understand why F = ma plus the position and velocity of all particles in a system are enough th know everything!

Thanks in advance
 

Answers and Replies

  • #2
Simon Bridge
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Q1. Well you also need to know the forces.
The rest follows automatically since classical mechanics is completely deterministic.

F=ma is the definition of what we mean by "force"... it's done that way because it is useful in predicting the outcome of experiments.
We do not always use the second derivative to define things. for eg p=mv

Q2.
Newton did not use Jerk in his equations because he didn't need to.

Basically, to find out where something is going to be, you need to know where it is now, how it is moving now, and how that motion is going to change later.
Isn't this common sense?

I suspect what you are missing is the understanding that the acceleration in F=ma is allowed to change over the trajectory of the particle.
So that F can be a very complicated function.
 
  • #3
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Can someone provide me a pratical exemple that shows how we can derive everything from a physical system given the position and velocity of every particle?
Knowing the orbital velocity of the planets and where they are right now is enough for us to calculate what the solar system will look like for many thousands of years into the future. Or at least it does if you keep in mind that...
Can someone better explain this to me? I can't understand why F = ma plus the position and velocity of all particles in a system are enough to know everything!
The "system" is defined by specifying what the forces acting on each particle will as a function of their positions and velocities. In this solar system example, we can use Newton's law to calculate the gravitational force on each planet and the sun as a function of their current position.

If we know the forces at any given moment, then we can calculate the accelerations. That, plus the velocities and positions is enough to calculate the positions and velocities a moment later; everything's velocity will change according to its acceleration and everything's position will change according to its velocity. And once we know the positions and velocities a moment later, we can repeat the process to get the positions and velocity the moment after that, and we can carry this on as far into the future as we please.
 
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Thanks a lot for the answers! It makes sense!
 

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