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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 let's 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 can't 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 canceled 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
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 let's 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 can't 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 canceled 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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