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John_Doe
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Why does F=dp/dt?
da willem said:the chain rule gives
Which is why the basic equations in classical mechanics PROPERLY stated (for the material system) should be be F=ma and conservation of mass (from which F=dp/dt trivially follows).Physics Monkey said:I would just point out that force has always been
[tex]
\vec{F} = \frac{d \vec{p}}{dt},
[/tex]
this is the way Newton wrote it in the Principia. The simplification that [tex] \vec{F} = m \vec{a} [/tex] can of course be made for most systems, but Newton did have it right from the beginning.
Also, the equation
[tex]
\vec{F} = \frac{d \vec{p}}{d t}
[/tex]
cannot be just simply differentiated to obtain
[tex]
\vec{F} = \frac{dm}{dt} \vec{v} + m \frac{d \vec{v}}{dt},
[/tex]
one simple way to see this is that the right hand side of the above equation isn't Galiliean invariant!
To add another way of looking at it is from the point of view of the action: [tex]S[/tex]John_Doe said:Why does F=dp/dt?
arildno said:Because:
1. It is readily verified that, within classical limits, mass does not vary with velocity.
2. Therefore, F=ma is NECESSARILY Galilean invariant (whether or not we have verified that a closed system has conserved mass) , whereas F=dp/dt requires the ADDITIONAL assumption for the closed system that mass is conserved in order to be Galilean invariant.
3. Thus, to regard F=ma and mass conservation as the basic equations for the closed /material system is simpler than F=dp/dt and mass conservation.
The relation F=dp/dt is not an equality but an identity. I.e. one does not derive this relation. It is a definition of F in terms of p. It used to be that it was a postulate and referred to as Newton's second law of motion but physicists have a difficult time when its referred to as a law since it is unclear how to measure F unless it is first defined. However this in itself opens up a whole can of worms since different people have very strong feelings about what I just said on both sides of the issue.John_Doe said:Why does F=dp/dt?
John_Doe said:If it were a definition, that would also logically imply that it were also an axiom