
Why does F = ma? - csantini
http://cognitivemedium.com/f-ma
======
KenoFischer
I'm not sure the question is particularly well posed, but I think the most
coherent answer you'd get from modern physics is symmetry. Restating Newton's
law's slightly, you basically get two points:

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.

~~~
lordnacho
This seems to lead to another question, which is why is the invariant of the
system what it is? Why are momentum (mv) and energy (0.5mv __2) conserved?

~~~
_Microft
A sibling comment to yours mentions Noether's theorem that links symmetries
(look up what symmetry means in this context if you're not sure) with
conserved quantities. Here are some examples:

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...](https://en.wikipedia.org/wiki/Charge_conservation#Connection_to_gauge_invariance)
for some details, even though I'm aware that it might be way too technical.

~~~
Filligree
It's interesting to note that, when you look at the universe as a whole,
energy is _not_ conserved. (It also isn't time- translation-symmetric.)

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.

------
knzhou
> Putting it in somewhat fuzzier terms, and at the risk of repeating myself: F
> = ma derives its power from the (implicit) assertion that there is a simple
> unversal force law that lets us figure out F for a particular configuration
> of matter. And so the configuration of matter completely determines the
> acceleration of a test particle. There is no a priori reason this ought to
> be true. It’s an absolutely incredible fact of nature.

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.

~~~
analog31
In a lot of problems, there is a "F = mv" as well, just not using the letter
m. A velocity dependent force is a "damping" force. Pushing an object through
some fluid medium will produce damping. There is also a "F = mx" which is a
spring return force. All three of these forces operating together drive the
(approximate) equation of motion for a lot of mechanical things, such as the
cone of a loudspeaker.

~~~
semi-extrinsic
Those F's aren't the same as the one in "F = ma" though. The one in "F = ma"
means "the (vector) sum of all forces on the object".

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.

------
roenxi
I liked KenoFisher's top level answer but I want to change the perspective
because I think momentum is a red herring.

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.

~~~
klodolph
This only makes sense in a relativistic context, but forces don't exist in
relativistic contexts.

So you can either have F = ma, or you can have GR, but you don't get to have
both at the same time.

~~~
selimthegrim
Wait, you can’t write the Lorentz force down?

~~~
cygx
You can:
[https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curve...](https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime#Lorentz_force_density)

------
tel
Isn't this derived from conservation of momentum? As I understand it, that
arises from a symmetry in spatial configuration: that physics behaves
identically along shifts in position.

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.

------
_Microft
Hijacking this thread in hope that some fellow physicist could chime in.

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?

~~~
knzhou
The electromagnetic field configuration does contribute energy/momentum. In
general, every field that interacts with the election field contributes: an
“electron” stands for a complicated joint excitation of the electron, photon,
W/Z boson, quark, gluon, ... quantum fields and what we call the
momentum/energy of the electron is just the total of that of all of these
fields.

------
dvt
> I find it astounding that a theory like quantum mechanics can have inside it
> another theory, an approximation, also extremely beautiful mathematically,
> but radically different. It’s like taking Bach, adding some noise, and
> getting the best of the Beatles out. I wish I understood better why this can
> happen.

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.

[0]
[https://en.wikipedia.org/wiki/Ehrenfest_theorem](https://en.wikipedia.org/wiki/Ehrenfest_theorem)

~~~
gus_massa
>> _It’s like taking Bach, adding some noise, and getting the best of the
Beatles out._

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.)

~~~
daxfohl
Better analogy would be chemicals adding up to life. Life adding up to
ecosystems. Bargaining adding up to economics. D flip flops adding up to the
internet. But all of this stuff is real.

~~~
gus_massa
I disagree. My idea is that with some common assumptions and simplifications,
QM looks like CM when the mass is big and other usual conditions. Like a real
piano can be under normal conditions simulated by an electric keyboard.

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.

~~~
daxfohl
Oh well, I'm more of a rolling stones fan anyway. (smiley face)

------
whatshisface
F = ma because if you walk outside and push things around that's what you will
notice is happening. Any other explanation including a derivation from QFT
would just be a different version of that, because whatever deeper underlying
theory you were using would have come from the same place as F = ma came from
to begin with.

------
ISL
We will never know why it does, even though it is a well-motivated prediction
of classical and modern theories.

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...](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.150801)

(I'd go dig up more citations on the subject, but we have a grant proposal due
in two days... :) )

------
theothermkn
'F = ma' is a special case of 'F = d/dt(mv)'. Expanding the latter via the
product rule, we get: F = m dv/dt + v dm/dt, where a=dv/dt and dm/dt is the
time rate of change of the mass of the object. This fuller form matters in,
for example, aerospace engineering, where the mass of the rocket is changing
as fuel is burned, so dm/dt is emphatically not 0.

I wonder if the philosophical questions asked in the fine article could have
been addressed more satisfyingly from that more general starting point?

------
andrepd
If you're interested, Landau & Lifschitz "Vol 1: Mechanics" has a derivation
of classical mechanics from purely physical arguments, from first principles.
F=ma is on page 9.

~~~
acqq
And that derivation seems to be known at least since 1788:

[https://en.wikipedia.org/wiki/Lagrangian_mechanics](https://en.wikipedia.org/wiki/Lagrangian_mechanics)

~~~
andrepd
Yes, but Landau derives that in a supremely elegant way, from purely physical
arguments.

~~~
acqq
I personally always prefer first historical approach, only then elegant
summaries. I am aware that some like just a distilled material. Still thanks
for mentioning Landau and Lifschitz book!

------
bobcostas55
As Poincaré put it, "Masses are co-efficients which it is found convenient to
introduce into calculations."

------
kmm
> [...] Two test particles with the same initial position and velocity, but
> different electric charges, can behave quite differently in the same
> electric field.

> 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.

[https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory](https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory)

------
ta1234567890
> the equation in Newton’s second law isn’t F = ma, but rather the more subtle
> statement that force is equal to the rate of change of momentum of a body

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.

~~~
raverbashing
The "AC" version of that equation is actually F=dp/dt (p = momentum)

~~~
ta1234567890
Thank you for the formula.

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?

~~~
ianopolous
Rockets

~~~
ta1234567890
Care to elaborate?

~~~
aerodude
Rockets have an exhaust with constant velocity, and they expel mass in the
form of fuel. It's this expulsion of mass that generates the force that
propels the rocket. Taking a force balance and assuming no external forces
(e.g. gravity), you can integrate to get the rocket equation, which tells us
how much fuel mass we need to reach a given velocity (the quantity we care
about when talking about stellar distances).

------
millstone
One interesting fact that has been missed here is the nature of time reversal.

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.

------
dboreham
Coincidentally on vacation I'm reading Leonard Suskind's "Special Relativity
and Classical Field Theory: The Theoretical Minimum", which (I think, if I'm
understanding correctly), goes into this question. Something about Gauge
Invariance..

[https://g.co/kgs/dygVQN](https://g.co/kgs/dygVQN)

------
whatshisface
> _A fun question: how does the universe change if the mass isn’t a scalar,
> but rather is a matrix, and so a = m-1F is the acceleration? What would this
> world look like? Is it plausible?_

This happens in our universe when the coordinate system is squished. In the
general curvilinear case, mass is a matrix.

~~~
knzhou
Not only that, but mass is a matrix for electrons moving in a crystal lattice,
even for normal coordinate systems. It's also true if you want F = ma to
continue to hold in special relativity, then m is a velocity-dependent matrix.

~~~
cygx
_It 's also true if you want F = ma to continue to hold in special relativity,
then m is a velocity-dependent 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.

------
bluesign
I think best way to look at this is m = F/a. Then mass is basically objects
resistance to force.

~~~
Anon84
This is a perfectly fine way of looking at it. But here's an interesting
question:

\- 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)?

~~~
saagarjha
Follow up question: is there an “electrical” inertia? Why does gravity have
only one relevant quantity but electromagnetic forces need two?

~~~
Anon84
Well, it really doesn't. The "problem" is that charge is always associated
with mass, with the charge generating electromagnetic fields and the mass
being responsible for the gravitational one

------
d--b
I personally would have gone with: because E=0.5 _m_ v^2

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.

~~~
earthicus
Here's a good explanation (it reduces to F=ma, of course!). This explanation
starts by answering a different question, then points out that energy is
analogous.

[https://physics.stackexchange.com/questions/67432/intuitive-...](https://physics.stackexchange.com/questions/67432/intuitive-
meaning-of-factor-2-in-formula-of-vertical-throw-max-height-
h-v2-2g/67433#67433)

------
octoboto
> _A fun question: how does the universe change if the mass isn’t a scalar,
> but rather is a matrix, and so a = m-1F is the acceleration? What would this
> world look like? Is it plausible?_

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

------
doggydogs94
One time in a physics class we were required to have a “cheat sheet” of
formulas for an exam. One student did not have a “cheat sheet” and was told
that he was required to have a cheat sheet. He grabbed a sheet of paper and
wrote F=ma. He aced the exam.

------
8bitsrule
Because Work = Force times Distance = Energy. Acceleration is the result of
energy added (by doing work) to a body with inertia (mass). Now if you ask
what is inertia: _that_ is the right (IMO unanswered) question!

------
jules
I don't understand the problem the author sees with the conventional
explanation. Yes, test particles of different charges behave differently, but
this in no way implies that the law should involve the third derivative of
position! The author says:

> 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.

------
ijiiijji1
Remember that F and a are vectors and m is a scalar.

~~~
chongli
Yes and that it’s more properly ΣF = ma. A lot of students forget that it’s
the sum of the force vectors, not just a single force, when they’re first
learning physics.

------
w0mbat
What confuses many people in high school is that F = ma but KE = 1/2 mv^2 (or
1/2 ma).

~~~
bermanoid
Acceleration does not equal velocity squared...

~~~
w0mbat
That would explain a lot. Our 80’s high school physics teacher taught us that
acceleration equals velocity squared, but he was prone to using vast over
simplifications to get through the material quickly.

~~~
elil17
That's not a vast oversimplification, it's completely unrelated to reality

