What Is The Meaning Of Newton's Laws

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Here are Newtons Laws

1. A particle at rest stays at rest or continues to move in a straight line at constant velocity unless acted on by a force.
2, Force is mass x acceleratoion
3 To every action there is an equal or opposite reaction.

What are they saying? Well 1 follows from 2 which is a definition. 3 is a statement about nature that can be tested but is equivalent to conservation of momentum. Generally it is assumed when using these laws what you are analysing resides in an inertial frame which has the property of being homogeneous in space and time (ie all points and times are equivalent as far as the laws of physics are concerned), and all directions are equivalent. Now from Noethers theorem this means momentum is automatically conserved. So on the surface the laws would seem vacuous. If you read Feynman's Lectures he argues they are testable - but except for law 3 they are not - doesn't matter what you do its circular. For example lets say you use a spring to apply a fixed force to test it. But how do you know what force it applies - somewhere alone the line you would have to have used F=MA to determine it.

So what is it saying - surely Feynman cant be wrong in saying it is a law with physical content? Well it is - but its not a law in the usual sense like say Coulombs Law which is an empirical statement. It is a law about how you should analyse classical mechanical problems. It says get thee to the forces. How do you test it? Well you see if you can use this definition to solve mechanical problems and you find you can and the answer is correct. So scientifically its valid. But you also find as the problems get more complicated applying the law gets harder and harder. If you are a great physicist like Feynman you can do it, and in fact when he studied advanced mechanics he did just that - but with great ingenuity and difficulty.

What has been found is that there is another formulation that is equivalent but easier to apply in complex situations - called the Lagrangian formulation. It says for any system you can find a function of velocity of the particles, position of the particles, and time, that when that function is integrated over the particles path it is a minimum. It called the principle of least action, and unlike Newtons is a law in the usual sense - you can directly verify it experimentally. The function not surprisingly is called the Lagrangian
.
It has other advantages as well:

1. Its existence is the assumption that goes into Noether's Theorem.
2. We know why it's true immediately - from Feynman's path integral approach to QM the only paths that exit here in the macro world are those of least action - the rest are cancelled by nearby paths.
3. It is the most powerful method in solving problems - when you have to be a real genius like Feynman to solve some problem using forces, with this method it's usually a lot easier..

So the actual basis of classical mechanics is QM. When you test if classical mechanics is true you are really testing if QM is true.

Thanks
Bill
 
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Answers and Replies

  • #2
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Here are Newtons Laws

1. A particle continues to move in a straight line unless acted on by a force.
2, Force is mass x acceleratoion
3 To every action there is an equal or opposite reaction.
Newton's Laws actually are

1. Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
2. The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
3. To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Together with the definition of motion (p=m·v) they define interactive forces - all three laws together, not only one of them.
 
  • #3
stevendaryl
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In my opinion, Newton's treatment of force doesn't say that a force is mass times acceleration. It says that mass times acceleration for a body is equal to the net force on the body. There can be forces that don't contribute to acceleration because they are canceled by other forces.

Maybe force can be thought of as "potential for acceleration".
 
  • #4
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Maybe force can be thought of as "potential for acceleration".
You are talking about internal stresses or strains. The net effect is still zero force because the acceleration is zero. But you make a good point - tension for example is best explained using the concept of force. Of course such is easily recovered from the Lagrangian formulation via the concept of generalized force.

Thanks
Bill
 
  • #5
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Maybe force can be thought of as "potential for acceleration".
I would reather see it as transfer of momentum - similar to the transfer of energy by work or heat. In the analogy to the first law of thermodynamics, momentum would be the state function and force the corresponding process function.
 
  • #6
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Together with the definition of motion (p=m·v) they define interactive forces - all three laws together, not only one of them.
Yes of course I was being a bit slack - I will modify the first law to be clearer.

Thanks
Bill
 
  • #7
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I would reather see it as transfer of momentum - similar to the transfer of energy by work or heat. In the analogy to the first law of thermodynamics, momentum would be the state function and force the corresponding process function.
I personally would rather see it taught after Lagrangian mechanics - but you need much more mathematical sophistication - so much so its impossible. I look at it as a consequence of the PLA. Although rate of change of momentum is not too bad (which is the definition of generalized force anyway - the derivative of generalized momentum - by Noether something is going on with that particle if its momentum changes - it is no longer a free particle and force is a good way to quantify that.

Thanks
Bill
 
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  • #8
stevendaryl
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This might be veering off-topic from @bhobba's point, but there is a whole field of Newtonian mechanics, which is static analysis of forces. There is no acceleration, and there's no momentum, but there are forces. The third law works for static situations, as well.

An alternate history of forces might measure forces in terms of how much they stretch or compress a spring in an equilibrium configuration. There could then be a theory of equal and opposite forces without ever mentioning acceleration or momentum. In this alternate history, it would be an empirical discovery, rather than a definition, that forces cause accelerations.
 
  • #9
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2, Force is mass x acceleratoion
Sorry for the nitpick, but this should be “Net force” and not just “Force”.
 
  • #10
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Sorry for the nitpick, but this should be “Net force” and not just “Force”.
That's fine.

Just being slack - which of course I should not be.

Thanks
Bill
 
  • #11
stevendaryl
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I agree with Bill that it is very cool how Feynman path integrals explains Lagrangian mechanics. The way Lagrangian mechanics is described is a little mystical-sounding: A particle going from point A at time t1 to point B at time t2 moves so as to extremize the action. It's hard to see how the particle manages to do this. Does it try all possible paths before deciding which one to take for real? It's also a little backwards, causally. How can some future event (ending up at B at time t2) influence what path a particle takes now? Of course, Lagrangian mechanics unlike Newtonian mechanics isn't intended to be a causal explanation of motion, it's just descriptive. But it always struck me as mysterious.
 
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