# Revisiting Newton's Second Law

I wonder why newton second law, define force as mass x acceleration, acceleration is the second time derivative of displacement, why he didn't define the force as mass x higher order time derivatives of displacement

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He did experiments and found that the second derivative of displacement is what worked.

I wonder why newton second law, define force as mass x acceleration, acceleration is the second time derivative of displacement, why he didn't define the force as mass x higher order time derivatives of displacement
A technical, but important point to note is that force is not defined as mass times acceleration, but instead as:

$$F = \frac {dp}{dt}$$

where ##p = mv## is the momentum. Note that this too, is a definition. If the mass is constant, then this becomes mass times acceleration. The reason he defined the force as this is just a choice. It is what works to describe what we see. In the physics framework (at least macroscopically), we make a distinction between what we see like speed, distance and acceleration, and the abstract part, like forces, energy and momentum. The goal of the abstract is to describe what we can see, and to this end we can come up with as many models, equations and so on. And occasionally, we find some very useful quantities, and we keep them around, and give them names. The history of kinetic energy is an interesting example. Momentum is another example; its definition is the simplest one (any factor could be multiplied or added to it) that is conserved, and it is a happy coincidence.
And because of momentum conservation, the force becomes a useful quantity. No matter what happens, you know the sum of the change in momenta is zero, so if you know the rate of change of one, you can know the rate of change of other things as well. Then you have a differential equation for each object that you can solve. Besides, with conserved quantities like momentum and energy, the actual value has no physical significance, only the change is important. Taking the derivative helps with that. And since momentum is conserved, you know that if the force on something is zero, that is, if the momentum is not changing, then the momentum is not changing: the velocity does not change. Those two statements sound trivial when you express them from momentum conservation, but they are in fact Newton's third and first law.

Then your question becomes, why is momentum defined the way it is, and why is it conserved? The question of why it is conserved can be answered by Noether's theorem, which basically states that because the laws of physics are the same everywhere in space, something must be conserved, and that something is momentum. Why is momentum defined the way it is? The motivation behind the quantity can be seen as a measure of how hard it is to stop an object: the more velocity an object has, the harder it is to stop, and ditto for mass. The importance of the velocity is evident, so the difficulty comes in the mass. And the reason why the mass is important is inertia. So basically, the force is defined as it is because of inertia and translational invariance.

Also, if you are not convinced about the fact that we choose the definitions, look at relativity. Momentum was redefined to preserve covariance of the law and its conservation. The force was not redefined, nor was the work done, so you really do get to choose what you want.

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Also, if you are not convinced about the fact that we choose the definitions, look at relativity. Momentum was redefined to preserve covariance of the law and its conservation. The force was not redefined, nor was the work done, so you really do get to choose what you want.

It seems to me that you could equally argue that if you redefine momentum and keep the same relationship between momentum and force, then you have redefined force.

It seems to me that you could equally argue that if you redefine momentum and keep the same relationship between momentum and force, then you have redefined force.
It depends on how you see it I guess. I look at the force as being defined by its relation to momentum, so I don't see it as being redefined. But if you were to compute the derivative you do end up with a different equation for the force. In a sense that is a redefinition.

Momentum was redefined to preserve covariance of the law and its conservation.

Not momentum but mass has been redefined in reltivity (in order to make it invariant again).

I was under the impression that momentum was also redefined as ##\gamma m_ov##. The rest mass was also redefined, but as the ratio of the energy in the rest frame to the speed of light squared because the idea of a rest mass was unnecessary in Newtonian mechanics. I guess you could say that momentum was defined as ##m_ov## in Newtonian mechanics.

I was under the impression that momentum was also redefined as ##\gamma m_ov##.

That's still the same momentum as definied in classical mechanics (here is a corresponding derivation). It's the meaning of the term "mass" which has been changed from ##\gamma m_o## to ##m_o## in relativity.

According to this paper, momentum is the one that's redefined: http://www.phys.ufl.edu/~acosta/phy2061/lectures/Relativity4.pdf.

It just states that "We can rewrite this momentum definition as follows:[...]". The formula for momentum changes but not momentum itself. It is also stated that the original formula p=m·v still applies in relativity if m is the relativistic mass. That means also according to this paper there is just a change of the definition of mass from relativistic mass (which is nothing else than the classical inertial mass under relativistic conditions) to rest mass. The definition of momentum remains unchanged.

It also states that "We need a new definition of momentum to retain the definition of force as a change in momentum", and that "In this definition of momentum, the mass m=m0 is the “rest mass”". It does change the definition of momentum ##m_o \frac {dx}{d \tau}##, and it does not use the relativistic mass. It mentions it in passing, but quickly points out that by using the relativistic mass, while momentum's definition is preserved, the definition of kinetic energy is not.
Though the whole thing of whether momentum is redefined or whether the mass is redefined is somewhat pointless. The point is, the way to calculate momentum changes, and the way to calculate the mass changes.

I"We need a new definition of momentum to retain the definition of force as a change in momentum", and that "In this definition of momentum, the mass m=m0 is the “rest mass”"

In my calculation (see link above) I demonstrated how the relativistic momentum results from the original definitions if Galilean trasformation is replaced by Lorentz transformation - without any redefinitions. The old definition is just rewritten using a new concept of mass. And even the new concept of mass is not a requirement in relativity. Relativistic mass would do the job as well. Using rest mass instead is just more comfortable.

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In my calculation (see link above) I demonstrated how the relativistic momentum results from the original definitions if Galilean transformation is replaced by Lorentz transformation - without any redefinitions. The old definition is just rewritten using a new concept of mass. And even the new concept of mass is not a requirement in relativity. Relativistic mass would do the job as well. Using rest mass instead is just more comfortable.

It is possible to keep the old definition of momentum, but you can also redefine the momentum instead, as the paper as shown. Your method is better though, and more natural to the eye. Redefining the momentum just like that seems a bit ad-hoc. There was a paper that used a Hamiltonian equation of motion (the derivative of the Hamiltonian with respect to the momentum is the velocity.) plus the fact that for kinetic energy to be conserved in two frames, the momentum must be conserved as well. The built the argument based on how the velocity transforms from frame to frame. I'll search for it and post the link when I find it

It is possible to keep the old definition of momentum, but you can also redefine the momentum instead, as the paper as shown.

I have a problem with the fact that there is no degree of freedom left for choices. My calculation shows that the momentum is already fixed by other conditions. Chosing the only possible solution isn't really a definition.

There was a paper that used a Hamiltonian equation of motion (the derivative of the Hamiltonian with respect to the momentum is the velocity.) plus the fact that for kinetic energy to be conserved in two frames, the momentum must be conserved as well.

That sounds much better. In this case momentum is not defined by an equation but by its's porperties. This is indeed another definition even though it is desigend to be in line with the original definition.

I get what you mean. I never liked this "momentum by definition" because it is ad hoc.

That sounds much better. In this case momentum is not defined by an equation but by its's porperties. This is indeed another definition even though it is desigend to be in line with the original definition.

I would think the definition is the same. The Hamiltonian depends on the momentum, where the momentum in the Hamiltonian's definition is defined as ##\frac {∂L}{∂q'} ## This definition of momentum is used in the paper, so it would be the same as Newtonian physics actually