# Is conservation laws are more fundamental than newton's second law?

Is conservation laws are more fundamental than newton's second law in Newtonian mechanics?
I know from the point of view of Noether's theorem conservation laws are more fundamental. But all the conservation laws can be derived from the F= ma equation. And from these conservation laws I cant derive newtons second law. So I am confused....Please give help

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Non-relativistically, E = p2/(2m), in which p is momentum. Therefore Sqr(E*2m) = p, in which Sqr is square root. You know that the time derivative of momentum is force (or you could postulate that)...and so force is probably the gradient of Sqr(E*2m) in which the scalar field would be based on scalar values of the velocity over how many dimensions you would prefer to take it over. To make things explicit, you could write the gradient out for momentum as Nabla(p(E(v))). What I just said might turn out to be bull**** because I just came up with it now, so you might want to wait for another response or check the validity of this one. In the case of potential energy, that already has a direct expression for force.

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Dale
Mentor
Is conservation laws are more fundamental than newton's second law in Newtonian mechanics?
I know from the point of view of Noether's theorem conservation laws are more fundamental. But all the conservation laws can be derived from the F= ma equation. And from these conservation laws I cant derive newtons second law. So I am confused....Please give help
You certainly can derive Newtons laws from conservation principles. If your generalized coordinates are the usual Cartesian coordinates of a system's particles then Lagrange's equation is Newton's 2nd law. For Newton's 3rd law, if your Lagrangian is invariant wrt spatial shifts then Newton's 3rd law results per Noether's theorem.

As far as which is more fundamental, you can derive either from the other, so it is hard to say. But the conservation principles seem to be better at describing the quantum world and they are coordinate-independent so they are more general.

Andy Resnick
Conservation laws are not derived from F=ma. IIRC, only if we restrict ourselves to certain 'allowed' forces (i.e. only conservative, non-dissipative forces) then can we 'derive' conservation laws. But the laws themselves in their full generality (and usefulness) are empricially obtained through experimentation.

Noether's theorem is important precisely because it links an empirically determined relationship to a very fundamental aspect of nature.

Conservation laws are not derived from F=ma.
I think we can derive conservation laws from F=ma.
1) when total force is zero then momentum is conserved. or for a closed system(where no external force present)
2)if we interpret F.ds as energy we can show PE+KE=constant. only thing we need is proper interpretation of potential energy and kinetic energy.
3) again for a closed system L=r*cross*P= angular momentum is conserved. we can show
show!!!

Is conservation laws are more fundamental than newton's second law in Newtonian mechanics?
I know from the point of view of Noether's theorem conservation laws are more fundamental. But all the conservation laws can be derived from the F= ma equation. And from these conservation laws I cant derive newtons second law. So I am confused....Please give help
The conservation laws are some integrals of motion (integrals of the Newton equations).
In the Newton mechanics of N particles there are 2N-1 integrals of motion, amongst which 7 ones are additive in particles (the total energy, momentum, angular momentum). In order to obtain the whole system of Newton equations for N particles, it is necessary to use (differentiate) all integrals of motion (2N-1 conservation laws).

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D H
Staff Emeritus
I think we can derive conservation laws from F=ma.
No, you can't.

when total force is zero then momentum is conserved. or for a closed system(where no external force present)
2)if we interpret F.ds as energy we can show PE+KE=constant. only thing we need is proper interpretation of potential energy and kinetic energy.
3) again for a closed system L=r*cross*P= angular momentum is conserved. we can show
show!!!
You are "proving" the hypothesis by assuming it to be true. That of course is not a valid proof.

One can derive the conservation laws for an isolated system from Newton's second law plus his third law. The weak form of Newton's third law (the force exerted by particle A on particle B and the force exerted by particle B on particle A are equal in magnitude to each other but opposite in direction) suffices to derive conservation of linear momentum. The strong form of Newton's third law (weak form plus the forces between particles are directed along the line connecting particles A and B) is required to derive conservation of angular momentum.

Regarding which is deeper? Newton's third law is not universally true. For example, magnetism can violate Newton's third law. The electromagnetic field itself has energy and momentum. Newtonian mechanics is only concerned with particles, not fields. The conservation laws still apply.

Andy Resnick
I think we can derive conservation laws from F=ma.
1) when total force is zero then momentum is conserved. or for a closed system(where no external force present)
2)if we interpret F.ds as energy we can show PE+KE=constant. only thing we need is proper interpretation of potential energy and kinetic energy.
3) again for a closed system L=r*cross*P= angular momentum is conserved. we can show
show!!!
You argument only holds for conservative forces. As evidence, I claim that you cannot account for friction in point (2).

Dale
Mentor
The strong form of Newton's third law (weak form plus the forces between particles are directed along the line connecting particles A and B) is required to derive conservation of angular momentum.
And (as I am sure you know) the strong form of Newton's 3rd law can also be derived using conservation provided that the Lagrangian is invariant under rotations as well as translations.