Is Newton's First Law a conservation law?

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Summary:

Is Newton's First Law a conservation law for motion (or velocity), comparable in a sense to conservation of energy or momentum?

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I'm thinking through a few basic things - hopefully in a new way. One thing that struck me is that momentum (mv) and energy (e.g. 0.5mv^2) can be conserved but velocity is not. For one thing, velocity is relative, of course.

I'm wondering whether there's a quantity a bit like velocity but not exactly. Call it "motion" or "movement". I'd want it to be agreed on by all observers in different reference frames. Could Newton #1 be regarded as some kind of conservation law?
 

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I'm thinking through a few basic things - hopefully in a new way. One thing that struck me is that momentum (mv) and energy (e.g. 0.5mv^2) can be conserved but velocity is not. For one thing, velocity is relative, of course.

I'm wondering whether there's a quantity a bit like velocity but not exactly. Call it "motion" or "movement". I'd want it to be agreed on by all observers in different reference frames. Could Newton #1 be regarded as some kind of conservation law?
Interesting question. No, I don't think it is inherently a conservation law by itself. However, I think it is more or less a consequence of the conservation of energy and momentum laws. If we think about it, both make sense. As you have said, Newton's 1st law states: an object will remain in uniform motion (current state of motion) when no net force acts on it (or words to that effect).

The described force can be denoted by an impulse for the momentum argument, or work done (for the energy method).

I could be overlooking something, but the only way for it to be a conservation law would be if it had another component which could be used to describe an object's motion which hadn't been covered by the other two.

Hope that is of some help.
 
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Ibix
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For the case of a single body with no external forces, conservation of momentum says ##m\vec{v}_{initial}=m\vec{v}_{final}##, which reduces to Newton's first law. So I'd say Newton's first law is a special case of conservation of momentum, presuming we're allowed to assume the conservation of mass.
 
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One thing that struck me is that momentum (mv) and energy (e.g. 0.5mv^2) can be conserved but velocity is not. For one thing, velocity is relative, of course.
Conserved and relative are not related/not mutually exclusive.
I'm wondering whether there's a quantity a bit like velocity but not exactly. Call it "motion" or "movement". I'd want it to be agreed on by all observers in different reference frames.
Acceleration.
 
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Could Newton #1 be regarded as some kind of conservation law?
In case force is expressed as gradient of potential energy, i.e. ##F=-\nabla U(r)##, the first law is written as
## \frac{d}{dt}(U+\frac{1}{2}mv^2)=0##,i.e. conservation of energy.

[edit] I mistake. The above was about the 2nd law.
 
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I'm thinking through a few basic things - hopefully in a new way. One thing that struck me is that momentum (mv) and energy (e.g. 0.5mv^2) can be conserved but velocity is not. For one thing, velocity is relative, of course.

I'm wondering whether there's a quantity a bit like velocity but not exactly. Call it "motion" or "movement". I'd want it to be agreed on by all observers in different reference frames. Could Newton #1 be regarded as some kind of conservation law?
On the face of it, the first law is just a special case of the second law with zero force and acceleration. But, you can think about the first law as defining the inertial reference frames in which the other laws hold.

For example:

http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node9.html
 
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Newton's first law is the Galileo inertia principle. This is, of course, the application of the law of conservation of momentum to a single body.

The world of Newtonian physics is defined by the Galileo principle of relativity, which differs from the principle of inertia.

Newton's first law is by no means a particular case of his second law. The co-operation of the first and second laws is illustrated by the fourth definition of Principia's first book. It is that if a body is at rest, or it is in a straight-line uniformly moving state, so this status can only change due to the impact of some force. However, this force will not remain in the body, and Newton's first law will work here. The second law, however, gives the possibility to calculate force, using definition IV.
 

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