Why does F = ma?
cognitivemedium.com
cognitivemedium.com
1. Momentum is conserved
2. All interactions between objects are mediated by forces (i.e. interactions exchange momentum).
So you may ask, ok, but why do these two things happen, which is is where symmetry comes in. If you require that your theory is invariant under translations in space and time, your theory must conserve momentum and energy. Then you ask, ok, so what kind of long-range interactions can we have between particles in a theory where energy and momentum are conserved and the answer turns out to be, well those that exchange momentum. For example, in the hypothetical from the blog post where interactions are mediated by exchanges of acceleration (forces are proportional to jerks), you end up with a universe where absent interactions, acceleration is constant. Why is this inconsistent with symmetry? Well, in such a universe, the acceleration of particles would be constant in the absence of interaction, so their kinetic energy would keep growing and growing, violating conservation of energy.
Energy is conserved because the laws of physics at two different times are the same.
Momentum is conserved because of translation invariance, as the laws of physics at two different points in space are the same.
There are others, like conservation of angular momentum because the laws of physics do not change under rotation.
Conservation of charge is because of gauge invariance of the electromagnetic field and might be the least understandable of the mentioned ones here. Check https://en.wikipedia.org/wiki/Charge_conservation#Connection... for some details, even though I'm aware that it might be way too technical.
It's been such a successful concept that you'll even find people denying that that can be the case. I'm one of them, somewhat -- I suspect it implies that we're missing part of the system, hence the universe as we know it isn't closed.
Consider releasing a rock and calculating how long it takes to hit the ground. In this system, the rock's momentum increases in time, and is therefore not conserved.
Consider a charged particle in an oscillating electric field. In this system, the particle oscillates, and so energy is not conserved.
What about Noether's theorem? In the first example the Lagrangian has a height dependence, and in the second one there's a time dependence. So the theorem does not apply.
Yet F=ma still works. So F=ma does not come from symmetry.
You might object that these quantities are conserved globally. But the point is that F=ma allows us to use model systems as affected by external forces; this is a very powerful simplifying technique.
m = P_S / c² * V_fohs
P_S and V_fohs do not exist in the standard model. The gravitational microcuviture of protons/neutrons cause changes in the P_S (static vacuum pressure) that therefor changes the mass. P_S is around 1.3 * 10^26 N/m².
This is from the Basic Structures of Matter - Supergravitation Unified Theory model, not the standard model.
A ball falling to earth has a time-independent Lagrangian because you neglect the earth's acceleration. The electron orbiting the hydrogen nucleus moves in a fixed potential for the same reason.
"Draw the boundaries correctly" and you can't do physics.
Such that things didn't just start moving on their own with no explanation. This is also why gravity was easier to approximate to them than heat. (Heat is symmetrical but hard to keep invariant, unlike motion.)
This is also why heat was considered an item (phlogiston / caloric theory) not a property of matter until quite late. (With Joule.) Or why the link between light and heat had to wait for so much longer.
Symmetries are actually much harder to pull off and verify. The big change there came with Maxwell or thereabouts.
Yup, this is totally correct. To say it yet another way, we evaluate scientific theories not by looking at the pieces in isolation, but how much explanatory power you get from all the pieces working together (penalized by the total complexity of those pieces). There's nothing mathematically inconsistent about defining "F = mv", it just makes F a less useful quantity.
Feynman actually had a remarkably similar discussion in his classic lectures:
> For example, if we were to choose to say that an object left to itself keeps its position and does not move, then when we see something drifting, we could say that must be due to a “gorce” — a gorce is the rate of change of position. Now we have a wonderful new law, everything stands still except when a gorce is acting. You see, that would be analogous to the above definition of force, and it would contain no information.
> The real content of Newton’s laws is this: that the force is supposed to have some independent properties, in addition to the law F = ma. [...] It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.
You can write down, as you say, a bunch of contributions like F_spring and F_damp etc. that can be functions of other variables like x, v etc.
"F = ma" then tells you that if you sum up all the forces (F_spring, F_damp etc) you have an expression for the acceleration.
This gives you an equation of motion that you can solve to find out how the object will move given some initial state.
Basically, if you know the initial position and velocity, you can compute acceleration at that time. Then you can figure out the position and velocity of the object a short time later, and recompute the acceleration using these new values. Then you can step forward again, and again, for as long as you want.
I like Feynman's analogy of physical laws to a chess board.
>"Every once in a while we get to peak at the board," 'but we don't really know the rules driving the game, we're just trying to figure them out.'
The starting point should be energy. Classical physics (and modern physics as far as I recall) determined that literally everything is linked by a single concept of energy. In my view energy is more real than anything else we experience and all our senses are for perceiving energy in different states. Matter is pooled energy, movement is energy, heat is energy (linked to movement), time is defined to some degree in terms of mass so it is linked in there too somehow. And we know energy is conserved because we have observed that everywhere.
Once we know energy is conserved the law of momentum makes a lot of sense - there have to be symmetrical laws that don't allow arbitrary creation of energy because that doesn't happen. And then F = ma turns up and it isn't so surprising for the same reason.
F = ma is in that sense not a fundamental law; it is a corollary of the conservation of energy. Mathematically there might have been other options, but that is the one that applies in this universe. There was always going to be a mass component in the formula because by observation energy is fundamentally linked to mass.
So you can either have F = ma, or you can have GR, but you don't get to have both at the same time.
Yes, they do; the difference in General Relativity is that gravity is not a force, but non-gravitational forces are still there.
We can imagine systems where energy is not conserved. Physicists work with these systems all the time. Electrical engineers consider the power coming from the outlet, not the entire (solar-sized) system which conserved energy.
In our EE's system, E is not conserved but F=ma still holds. So F=ma is true even if energy is not conserved.
Well; no. Energy is conserved. Just because the electrical engineer is ignoring that fact doesn't stop it being important. Same with the physicists.
F = ma is a very convenient equation. But energy conservation is a simpler, more fundamental concept and forces are a mathematical artefact of that which happen to be easy to work with.
I'm well beyond anything I'm familiar with, but it feels like these more fundamental relations should be where meaning of equations like F=ma arise. If momentum is conserved due to symmetry of position, then forces applied by a field are exactly what disrupt that symmetry and thus are exactly what invoke changes in the conserved quantity?
I hope for explanations like that to bear more fruit because they arise from some pretty undeniable facts of reality: there aren't privileged positions in space except for how forces exist in some configuration, there aren't privileged moments in time except for how events occur along some timeline.
I'm physicist myself so maybe I should know this or be able to figure it out myself but here's the question:
Where is kinetic or rotational energy 'stored'?
Is it eventually the transformation of fields that gives rise to kinetic energy? For example an electron whose electric field is not a radially symmetric must be a moving electron and the difference in energy between the field configurations (simple radially symmetric electric field on one hand and the mix of electric and magnetic fields on the other) is exactly the kinetic energy it has?
https://en.m.wikipedia.org/wiki/Abraham–Minkowski_controvers...
This isn't really the case. Author should've probably mentioned the Ehrenfest theorem[0]. It's not exactly accurate to say that Newtonian mechanics is "embedded" in quantum (or rather Hamiltonian) mechanics. In fact, we need to do some "artificial" re-jigging to make quantum mechanics work at Newtonian scales.
It's a very bad analogy. It's difficult to find a good one. Let's try anyway: Imagine that QM is a real piano and CM is how you expect a piano to sound in normal conditions.
If you press a C key, you get a C sound. You can modify the sound releasing the key or keeping it pressed, or pressing the pedals. There are some effect where you press another keys so the sound of the original string make the other strings move and modify the sound. You can hit the wood part of the piano to make some drum like sounds.
Most of this can be simulate in an electric keyboard and get an accurate piano-like sound. This is like Classic Mechanics.
Quantum Mechanics is like the real piano. You can open or close the top. You can fill the piano with a different gas to make the sound slightly different. Or fill it with water or liquid nitrogen. Use drumstick to hit the strings directly. Use a hammer to hit the wood walls. These create a more wide set of sounds that are nor reproducible in a electric keyboard. An electric keyboard is only an approximation of an idealized real piano.
(An experiment where the waviness of the particles is the most important part is like dropping a piano from a 8th floor to hear the sound of the crash, or using the piano as firewood to hear the crispy sound of fire in a winter night. I'm not happy with these examples because they are too destructive, but I wanted too examples of sounds that are very far away from the sounds you expect in an electric keyboard.)
(For some reason, musicians do more weird destructive acts with guitar than with pianos. Perhaps it would be a better instrument for the analogy. But it is more difficult to make an idealized model of a guitar. A piano is almost discrete.)
Life is like an emergent phenomena based in chemistry. In some special cases, the chemistry is very complicated and it's better to have special name for this situation and call it life.
Anyway, the problem with analogies is that each one thinks that their own one are better.
It is possible for us to test whether or F=ma, though, and so far, it (with relativistic corrections) checks out.
An example of this sort of test from our research group: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98...
(I'd go dig up more citations on the subject, but we have a grant proposal due in two days... :) )
I wonder if the philosophical questions asked in the fine article could have been addressed more satisfyingly from that more general starting point?
> One possible response is to say “oh, maybe our notion of force should really be something like F = mj, where j is the jerk, i.e., the third derivative of position”.
> I’ve never worked it out in detail, but wouldn’t be surprised if such an approach can be made to work.
It might not be exactly what he's looking for, but in Kaluza-Klein theory, charge is related to the velocity of the particle in an unseen 5th dimension. Through some miracle, if you work out general relativity with one dimension extra, the resulting theory bears a striking resemblance to general relativity in the usual four dimensions plus electromagnetism. If you then assume the 5th dimension is extremely small, which would explain why we can't see it, you get quantization of charge for free.
This is probably the most insightful bit and points directly to the potential of fiddling with the equation.
Drawing an analogy from electricity, F=ma is in a way the "DC version/expression" of a force, in which m and a are constant and F is a fixed quantity. However we can write an "AC version/expression" of the equation as F(t)=m(t)*a(t), in which the average of F(t)=F.
In a way we tend to just be content with dealing with averages instead of looking at the detail and seeing how it varies over time.
There's a lot of potential in looking at the variation. Especially as technology enables it through high speed cameras and instruments with higher sampling rates/resolution.
Do you know of example advances that have been made or problems that have been solved as a consequence of applying the concept implied in that formula?
F=ma. Switch on the time reversal machine: t becomes -t. Positions stay the same, but velocities reverse, because dx/d-t -> -dx/dt.
BUT accelerations stay the same! dx/d^2t gets a double negative, which is just 1. Gravity attracts in the future and the past. Throw a ball up and it comes back down; reverse the camera and it looks the same.
This means that (handwaving) forces are independent of time. And this in turn leads to a time-independent construction of forces - how about as the gradient in space (NOT time) of a potential.
This I think is a path to partial insight. A particle moves in a potential, and we can calculate the quantity 'a' from the purely-spatial gradient at the particle's position, without reference to time at all. And we can do this because 'a' is an even derivative of position.
This is a crappy argument; at best it argues that it's nicer that f=ma instead of f=mv or f=ma'. Fourth or sixth derivatives are not considered. Still, understanding dynamics as governed by position only is compelling.
This happens in our universe when the coordinate system is squished. In the general curvilinear case, mass is a matrix.
Not if you describe things in terms of four-forces and four-accelerations, which restores the simple proportionality. However, the spatial component of 4-acceleration is not necessarily parallel to 3-acceleration.
- We know that Force is proportional to acceleration. Let's call the proportionality constant, m_1=F/a.
- We also know that objects are attracted to the Earth with a proportionality constant: m_2= F r^2/(GM)
and we can certainly set:
- F == m_1 a == G m_2 M/r^2
to calculate the acceleration of the gravitational force. But why should m_2==m_1?
In other words, why is "inertial mass" (m_1) the same as gravitational mass (m_2)?
In a way, it's more like
m·(a - a_grav) = F_nongravIn some sense, you could say that the ground is accelerating upward at a constant 9.8 m/s². When we feel weight, that's just regular inertia, pushing back against the ground's acceleration.
(I'm not sure to what extent this follows the mainstream interpretation of Einstein's equivalence principle. It at least requires a different definition of "acceleration", given that the Earth's radius is not increasing.)
And by that i mean that energy is a much more intuitive concept than force.
Why is E proportional to v^2? Now THAT’s an interesting question.
Hence v^2 is hiding the complexity of density vs surface area (the law was originally derived by observing steel ball impact craters on a bed of sand vs the height).
0.5 is because only one side hits.
EDIT: And what surface area? Well General Relativity does like to mash up the concept of objects affecting space time due to having energy...
https://physics.stackexchange.com/questions/67432/intuitive-...
m is only a scalar for particles though? For a general rigid body mass is a matrix. Unless they specifically mean for translational degrees of freedom, in which case I think that would probably break symmetry
> I’ve never worked it out in detail, but wouldn’t be surprised if such an approach can be made to work. Essentially, it’d make acceleration into a free (possibly constrained) parameter of the particle, rather than something completely determined by the distribution of matter and fields. That free parameter would implicitly contain what (in the conventional approach) we think of as the charge information. Indeed, the new equations of motion would have a conserved quantity, corresponding to the charge. But the resulting force laws would be quite a bit uglier.
I don't see how this could be true. If it is, I'd indeed like to see it worked out.
Particles with different charges still follow F = ma. The force is different for different particles, sure, but the evolution of the position of the particle still follows a second order differential equation and not a third order one. Furthermore, going to a third order equation doesn't even solve the problem that the F is different for particles of different charge, that problem is still there the same as before. I don't think is a problem at all, by the way: the forces always depend on the particular situation we're considering, including on the mass, charge, and so on, and even more so for compound objects.
The question "Why does F = ma?" is a good question though. The conventional explanation only explains why classical mechanics is governed by a second order differential equation. It does not explain why F = ma is the right equation. If you interpret F as a general function of the state of the system then indeed F = ma doesn't tell you anything, because by picking the appropriate F you can get any second order differential equation. However, if we already have some prior idea of what force and mass is, which we do, then it's not clear why it should be this differential equation. We do have an intuitive idea of what force is: if you hang 1kg on a rope then the rope pulls with some amount of force, and if you put 2kg then it pulls with double the force. Similarly, if you stretch a spring by 3cm you have some amount of force, and if you stretch two of those springs simultaneously you have double the force. You could relate the force in the spring with force by a weight by finding the amount of weight you need to stretch the spring by that 3cm. We can get a definition of force by picking some reference force as being 1 unit of force, and we can similarly come up with an experimental definition of mass.
If we accept such a definition then F = ma has physical meaning. For instance, it says that if you double the force then you double the acceleration. That is, if you pull an object with two springs then it accelerates twice as fast as when you pull it with one spring, if you stretch the springs by the same amount in both cases. This is a nontrivial fact about the physical world. The law F = ma also tells you that if you apply a constant force then the velocity increases linearly. That is, if you pull an object with a spring and keep the spring stretched by 3cm, then if the object accelerates to speed v in one second then it will accelerate to 2v in two seconds. This is a nontrivial fact about the physical world too.
Are you sure you just didn't misremember it?