Position and speed are enough in a physical system?

In summary, classical mechanics is a deterministic system where the state of a physical system and its future evolution can be determined by the instantaneous locations and velocities of its constituent particles. This is due to the fact that by knowing the forces acting on each particle, we can calculate the accelerations, which combined with the positions and velocities, can predict the future state of the system. This is why Newton did not use jerk in his equations and instead only used the second derivative, as it is sufficient in making accurate predictions.
  • #1
DarkFalz
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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/qu...ivatives-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
 
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  • #2
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
DarkFalz said:
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.
 
  • #4
Thanks a lot for the answers! It makes sense!
 
  • #5


I can provide some insights on the questions raised in the content.

Firstly, the statement that "the state of a physical system and its future evolution is fully determined by the instantaneous locations and velocities of its constituent particles" is referring to the principles of classical mechanics. In classical mechanics, the behavior of particles is described by Newton's laws of motion, which state that the acceleration of a particle is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is described by the equation F=ma. In this context, the position and velocity of particles are enough to determine their future motion, as long as the forces acting on them are known.

To answer the first question, a practical example of how we can derive everything from a physical system given the position and velocity of every particle is the motion of planets in our solar system. The positions and velocities of all the planets can be used to accurately predict their future trajectories and behavior, as long as the gravitational forces between them are taken into account.

Now, onto the concept of jerk. Jerk is the third derivative of position and is a measure of how quickly the acceleration of a particle is changing. In classical mechanics, jerk is not considered because it is not necessary to describe the behavior of particles. Newton's second law, F=ma, already takes into account the change in acceleration over time. Including jerk would only add complexity to the equations without providing any additional insights.

To further explain this, think of it in terms of initial conditions. As mentioned in the content, to solve a second-order differential equation (such as Newton's second law), we only need two initial conditions - the position and velocity of the particle. This means that if we know the position and velocity of a particle at a given time, we can use Newton's laws to determine its position and velocity at any other time. Similarly, if we know the position, velocity, and acceleration of a particle at a given time, we can use Newton's laws to determine its position, velocity, and acceleration at any other time. Including jerk in the equations would not add any new information, as it can be derived from the previous three variables.

In summary, the position and velocity of particles are enough to determine their future behavior in classical mechanics. Jerk is not included in the equations because it is not necessary to describe the behavior of particles, and can be derived from the previous variables. I hope this helps to clarify
 

1. What does it mean for position and speed to be enough in a physical system?

When we say that position and speed are enough in a physical system, it means that these two variables are sufficient to describe the state and behavior of the system at any given time. This is typically the case for simple systems, such as a ball rolling down a hill, where knowing its position and speed is enough to predict its future motion.

2. Are there any physical systems where position and speed are not enough?

Yes, there are more complex physical systems where position and speed alone are not enough to fully describe the system. This is often the case in systems with multiple interacting components, such as a pendulum with multiple moving parts. In these cases, additional variables such as acceleration and angular velocity may be needed to fully understand the system.

3. How do position and speed relate to each other in a physical system?

In a physical system, position and speed are closely related. Position refers to the location of an object in space, while speed refers to the rate at which that object is moving. In most cases, an increase in speed will result in a change in position, and vice versa. However, this relationship can become more complex in systems with acceleration, where changes in speed are not constant.

4. What factors can affect the accuracy of using position and speed in a physical system?

There are several factors that can affect the accuracy of using position and speed to describe a physical system. These include external forces acting on the system, such as friction or air resistance, as well as potential errors in measurements of position and speed. Additionally, the complexity of the system itself can also impact the accuracy of using these variables.

5. Can position and speed be used to predict the future behavior of a physical system?

Yes, in many cases, position and speed can be used to predict the future behavior of a physical system. This is because these variables provide important information about the current state of the system, which can be used to make predictions using mathematical models. However, the accuracy of these predictions will depend on the complexity of the system and any external factors that may affect its behavior.

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