# What is the fundamental definition of force?

As I pick up more physics I see that the definitions of force commonly provided in books and classrooms are misleading.

• "A force is a push or pull." This seems to be a "correct" definition but it doesn't provide enough information.

• "A force is the influence of one body on another." This is not sufficient because as other people have pointed out to me, force is more so the relationship between two bodies as opposed to how one acts on another. This is more evident with forces such as electricity and gravity.

• "$$\vec{F} = m \cdot \vec{a}$$." My understanding is that this is not a mathematical definition, but rather a scientific observation. Rigorous application of the scientific method led us to conclude that the relationship between force and acceleration is proportional, and the constant of proportionality is the mass of the given object. It's not a definition in the sense that we define velocity as displacement over time.

Can someone please provide an intuitive, natural definition which describes the inherent behavior between objects/bodies in the physical world? I understand that there are many different kinds of forces but since we call them all "forces" there must be a good way of defining all of them in a singular manner.

• Just because different things are referred to by the same word doesn't mean there's one definition that works for them all at once. In a car crash, the guy with the right-wing bumper sticker in the right lane might have the right of way but didn't signal right. In this case, what is the true, natural definition of "right"? Commented Feb 28, 2022 at 20:48
• It is quite rare for anything to have a single and universally accepted definition in any hard field. You always have a wide range of different definitions with overlapping regimes of validity and different degrees of usefulness and precision. Commented Feb 28, 2022 at 20:52
• Rather than looking for a definition, understand that all of physics is models (conceptual + formal descriptions) of observed phenomena. It has been observed that objects interact, sometimes over distance, causing each other to move; and that there's directionality to these influences. These interactions have then been termed "forces", have been modeled as vector quantities, and have then been related to other quantities and concepts within some theoretical framework. All that together is a complete description of what a force is. So, any concise def. has to be to some extent a summary. Commented Feb 28, 2022 at 21:14
• Related: Are Newton's "laws" of motion laws or definitions of force and mass? and links therein. Commented Mar 1, 2022 at 8:43
• I would say that the most basic definition of a force is the exchange of bosons between two fermions. I don't know how much that helps with the intuition about forces, but at least is an explanation for any force in physics, which is what I believe you are looking for. Commented Mar 4, 2022 at 16:11

## In Newtonian Mechanics

In Newtonian mechanics, a force is a mathematical vector we prescribe onto a model of a physical system by declaring a force law.

In other words, it's an intermediate mathematical gadget we invoke to do calculations in our models. It is invoked between the inputs (initial conditions) and outputs (predictions) of data but it is never measured directly (time, position, velocity, etc. are what are ultimately recorded directly).

To put it more bluntly, it is a primitive notion that can't be reduced further unless you step outside Newtonian mechanics. A force is supposed to be a mathematical concretization of what we intuitively regard as a "push or a pull" but there isn't going to be a perfect correspondence between the notion of forces and "pushes or pulls." It is only a rigorous refinement of those intuitive notions.

This is similar to how the concepts of a line or a point used to be primitive concepts in Euclidean geometry until the advent of real analysis and set theory. Prior to the widespread adoption of coordinates, the notion of a line was a mathematical primitive that corresponded to our intuitive notion of lines. The best we could do isn't to give a direct definition, but to give an axiomatic characterization of what we consider a line to be and work from there.

This is also similar to how the wavefunction is invoked as a mathematical gadget to do calculations for models of quantum systems; the wavefunction is also invoked between inputs and outputs but it is never directly measured. (Of course, I should say that unlike forces and lines, there is no intuitive thing in the real world that we can point to that the wavefunction corresponds to, which makes the situation in quantum mechanics more difficult to deal with. Moreover, while the concept of a force has a direct correspondence with position measurements through a spring, balance, or a general force gauge, the concept of a wavefunction doesn't have a direct correspondence with any specific measurement unless you take an ensemble of identically prepared systems.)

Consider the example below.

Example. Suppose I want to model a binary star system. I model the two stars as point objects with masses $$m_{A}$$ and $$m_{B}$$, and then I appeal to Newton's law of universal gravitation to specify the force as $$\vec{F}_{\text{A on B}} = -\frac{Gm_{A}m_{B}}{r^{2}}\hat{r}$$ where $$\vec{r}$$ is the vector from star $$A$$ to star $$B$$. This is something I put into my model, because this law has been successful for astronomical predictions.

Another example is given below.

Example. Suppose I want to model a harmonic oscillator put into a fluid with some drag. Then I postulate two force laws: the spring force $$\vec{F} = -k(\vec{x} - \vec{x}_{0}),$$ and the linear drag force $$\vec{F} = -b\vec{v}$$ where $$b, k$$ are some positive constants.

Some other notes:

• Something else that is important to understand is that neither Newton's first nor second law are used to define what is a force. Their roles are merely to relate force to motion. It's the force law specific to the situation that specifies the force, and then Newton's laws relate it to motion. Newton's first and second laws are not definitions of force so much as they are axiomatic characterizations of the relationship between force and motion. There is a subtle difference, because at no point do we say "a force is defined as blah-blah-blah" in either of the laws. The role of Newton's first and second laws are to relate force to the motion of objects, and in the process of doing this they elucidate what it means for a force to be "a push or a pull" or to be "an influence of one body on another."
• Newton's third law is different from the other two laws, because unlike the first two laws the third law gives a constraint on what the possible force laws (which are the things that specify what the force is in a given scenario) there can be. In many cases, we actually ignore this law to simplify our models (for example when we consider a spring attached to a wall, we simplify our scenario by ignoring the fact that the motion of the spring imparts some momentum to the Earth).
• Some forces are "more fundamental" in the sense that we can derive other forces from the more fundamental ones. For example, the spring and drag forces come from more elementary forces that act on the molecules of the substances.
• If you come across a new situation that no one else has analyzed, you will have to take apart the situation and see what mechanisms are at play that you are familiar with. From this you can deduce the force laws at play. The force laws can come from other theories such as electromagnetism where force is defined by the electric and magnetic fields (which are yet another slew of mathematical gadgets we invoke). In the worst case scenario where you can't take the situation apart, you guess the force law and empirically test whether or not your guess leads to correct predictions.

## In Lagrangian Mechanics

In different theories, we start out with different primitive notions. If we start with Newtonian mechanics, then mathematically speaking force is going to have to be a primitive notion. If we start with Lagrangian mechanics, then the Lagrangian $$\mathcal{L}(q, \dot{q}, t)$$ will be the primitive notion, and force will be defined as $$F_{i} = \frac{\partial\mathcal{L}}{\partial q_{i}}.$$ For $$\mathcal{L} = T-U$$, force ends up being defined as the negative gradient of potential energy: $$\vec{F} = -\nabla U$$.

## Other Theories

Note that outside of any specific theory, the word "force" doesn't have a rigorous definition, and it is mostly an intuitive concept. Also, the word "force" can be attached to different things in different theories (so in the broader sense it's an evolving concept as we figure out more things about the universe). In quantum field theory and general relativity, all Newtonian forces emerge out of more fundamental effects. We can choose to refer to those effects as forces or we can choose to say none of them are forces; the difference becomes a matter of semantics at that point.

In GR, the gravitational force emerges out of the dynamics of the motion of bodies in curved spacetime. In the Newtonian limit (objects moving slowly, weak gravitational field, relatively static gravitational field), Einstein's equation reduces as $$G_{\mu\nu} = \frac{8\pi G}{c^{4}} T_{\mu\nu}\rightarrow \nabla^{2}\phi = 4\pi G\rho_{\text{mass}}$$ and the geodesic equation reduces as $$\frac{{d}^2x^\mu}{{d}\tau^2}+\Gamma^\mu_{\nu\rho}\frac{{d}x^\nu}{{d}\tau}\frac{{d}x^\rho}{{d}\tau}=0 \rightarrow \frac{d^{2}\vec{x}}{dt^{2}} = -\nabla\phi.$$ If $$m$$ is the mass of the object being affected by gravity, then we can choose to call $$m\cdot -\nabla\phi$$ as the force $$\vec{F}$$ in Newtonian mechanics. By multiplying both sides of the last equation by the mass $$m$$ of the object, we recover the usual Newtonian formalism.

In QFT, three of the four fundamental interactions come down to an exchange of bosons between fermion particles, and the last interaction, gravity, is hypothesized to have a corresponding graviton particle (of course the nature or lack thereof of quantum gravity is currently outside the scope of experimental testing).

• Best answer so far IMO. "Time, position, velocity, etc are what are ultimately recorded directly" not sure about this statement: can we directly measure velocity? Are those 3 things the only things that can be "directly" measured? Commented Feb 28, 2022 at 22:26
• @Quillo I'm not 100% sure. I've seen people argue that time and position might be the only "directly" measurable things. In any case, as far as I can tell, every measure of force relies on a measure of something else. Maybe the feeling of pressure (force divided by area) seems like it doesn't involve position, but actually that occurs because our nerves send actual electrical signals which have to be carried by something... so even that involves changes in position. Commented Feb 28, 2022 at 22:46
• What is the force law for the normal force acting on a body from the ground? Sometimes forces exist without an explicit definition of their magnitude, but as enfocers of constraints. Commented Mar 1, 2022 at 13:30
• @Quillo One minor addendum is that I think we have to be careful in what we mean. For example, sometimes when people say "gravity" they may refer to the observable phenomena that things fall down. This is obviously not a mathematical gadget. Other times, when people say "gravity" they may refer to the force vector in Newtonian mechanics or to a term in a Lagrangian. Clearly, these two have different meanings. The same applies to the other forces/interactions. Commented Mar 2, 2022 at 16:11
• @MaximalIdeal I totally agree. I was just replying to a previous comment of ACuriousMind by noticing that the definition of "force" as "gadget" also applies perfectly to the classical concept as well as to "fundamental interactions". Commented Mar 2, 2022 at 16:18

"A force is a push or pull." This seems to be a "correct" definition but it doesn't provide enough information.

That is the most commonly cited qualitative definition. It's broader than using Newton's 2nd law since, as discussed below, Newton's 2nd law only addresses the influence of a net force. A force (push or pull) does not require that there is an influence.

Insofar as whether or not it provides enough information it depends on what kind of information you are looking for.

"$$\mathbf F = m \mathbf a$$." My understanding is that this is not a mathematical definition, but rather a scientific observation.

Newton's 2nd law provides information on what a force does. But if you are looking for a better mathematical definition of the effect of a force, I think you are better off defining the effect of a net force as the change in momentum of an object, or

$$F_{net}=\frac{dp}{dt}$$

where, for the case of constant mass,

$$\frac{dp}{dt}=m\frac{dv}{dt}=ma$$

The reason I believe this is a better mathematical description of the effect of a force is that conservation of momentum is one of the fundamental laws of physics.

The emphasis is on net force, because though "pushing or pulling" is a force, there may be no effect unless there is a net force. I can push and pull on a wall all day, but if it doesn't move (cause a change in momentum) my force has no effect (at least, macroscopically) on the wall.

"A force is the influence of one body on another." This is not sufficient because as other people have pointed out to me, force is more so the relationship between two bodies as opposed to how one acts on another. This is more evident with forces such as electricity and gravity.

I have a few issues of what you have been told here. For one thing, the influence may due to contact between bodies, or the influence may be due to a field between the two bodies. But the main reason not to define force as "the influence of one body on another", in my view, is as I said above, a force does not necessarily influence a body (read rigid body) unless it is a net force.

I'm actually more concerned with being accurate than being precise. Would it be fair to say that this definition applies to all forces in physics? "A force is a push or pull resulting from an object's interaction with another object."

I would say the "push or pull" definition applies at least to two of the four fundamental forces, i.e., the gravitational and electromagnetic force. I'm not so sure in the case of the other two, the strong and weak forces. As far as your original statement

I understand that there are many different kinds of forces but since we call them all "forces" there must be a good way of defining all of them in a singular manner.

That, of course is the Holy Grail. Gravity still has not been combined with the other three.

Hope this helps.

• I'm actually more concerned with being accurate than being precise. Would it be fair to say that this definition applies to all forces in physics? "A force is a push or pull resulting from an object's interaction with another object." Commented Feb 28, 2022 at 21:36
• @EthanDandelion I have revised my answer to respond to your follow up question. Commented Feb 28, 2022 at 21:49

I think that force has at least $$4$$ layers of meaning.

The primary meaning is an intensive quantity, something that we feel with our muscles, mainly when pushing or pulling. As such it is not measurable, because even if we can say that force A is bigger that B, it is not possible to precise how much.

In order to measure the force and treat it as an extensive quantity, we use the Hooke's law to make load cells, strain gages, and other devices. That is the second layer.

The discovery that net force is proportional to acceleration leads to the third level. As that is a more universal property than elasticity (which can have a short range, depending on the material), this discovery is 'promoted' to the standard way to measure the net force. And if a load cell is not perfectly linear with acceleration for a given mass, the Newton's second law prevails, and it is a way to fine tuning the degree of non-linearity of the load cell.

The fourth level derives from the observation that sometimes acceleration of particles is a function of the position in a system of coordinates, and for such cases, the (conservative) force can be defined as minus the gradient of a scalar function. In that way for example , even the force between $$2$$ protons of a $$H_2$$ molecule can be known as a function of their momentarily distance.

I happen to like Aristotle's definition although he didn't use the term force. Essentially, force is that which causes change. More precisely, he wrote in his Physics:

... anything which can cause change must cause something to be changed and it must be something that can be changed. Similarly, what can be changed must be changed by something and it must be something that has the ability to cause change ... when something changes, it inevitably does so in respect of substance, quantity, quality or place ... the upshot is that there are as many kinds of change as there are categories of being.

$$200^b26$$

His categories of being are four:

• actual existence and their change is 'coming to be' and 'passing away'.

• the number of things and their change is an increase or decrease in number. He means here integral number, like for example the number of atoms.

• quality, these are continuous things such as length or mass and their change is what he terms alteration - their continuous increase or decrease

• place, this is position and change here is just change of position, that is motion.

Thus force is that which can cause things to come to be, like particles coming into existence; or to pass away, like particles annihilating; and such forces obviously changes the number of particles, either their increase or decrease; more, forces are what causes change in volume, say pressure.

For Aristotle, the world is a network of forces inhering in matter and acting on matter and thus causing change in substance, number, quality and place.

It's worth seeing how Aristotle's definition of force stacks up against the classical definition, that of Newton. This is usually expressed symbolically as $$F=ma$$. But this is not what Newton wrote in his Principia, what he actually wrote was:

The alteration of motion is ever proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed.

Obviously, Newton's definition is much narrower than that of Aristotle's. He focuses only on motion. Thus his term 'motive force', a force that causes change in motion. The crucial question is whether Newton's law a specialisation of Aristotle's? Well we see a 'motive force is impressed' and this causes an 'alteration in motion'. Alteration is obviously change and change is what we are looking for when characterising a force according to Aristotle. And Aristotle does specify change in place as one of the kinds of changes possible. However, here Newton is not talking about change in spatial position but change in velocity. But equally soundly, we argue that velocities constitute a space, the space of velocities. So, yes, it does fit. Of course this is really an extension of Aristotle's definition as did not concieve of a space of velocities; however, he left his theory open to extension because although he characterised four main categories of being, he recognised that there were other more specialised senses.

Whilst Arostotle's law is broader, we see that Newton's law is quantitative, it says the change of motion is 'proportional' to the motive force as well as specifying the direction of change. Aristotle's definition is qualitative and as he said himself, one can become more precise as this law is specialised to more specific domains, as is here by Newton.

It's also worth noting that Newton says 'impressed' and this means that the force should act by contact. In fact, Newton felt philosophically that all forces should act by contact and this is why he understood his theory of gravity to be incomplete since it had forces acting at a distance. Newton doesn't say why forces should act by contact but it's likely the original source was Aristotle. In fact, he wrote:

Everything that cause change is changed ... as long it is capable of changing ... For to act on something changeable, in so far as it is changeable, is precisely to change it, and it takes contact to do this, so the agent of change is acted upon at the same time.

$$202^a3$$

Thus forces take 'contact' to act on and cause change. In Newton's language, they need to be 'impressed' upon. But more, we also see that the preceding passage is a qualitative statement of Newton's third law:

To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each another are always equal and directed to contrary parts.

Again, Newton provided the crucial quantifying information: these two forces are equal and acting in opposite directions.

Finally, I'd like to add a few words about what Aristotle meant by change. Change is familiar and ever since the discovery of calculus it has been straight-forward to model. Aristotle thought otherwise. He struggled to characterise change and his struggle had nothing to do with the lack of a suitable calculus. What he was stuck on was ontology. It is easier to say what something is or not. But change, which ontological category does that lie in? Aristotle himself said:

Also, the process of change does seem to be an actuality but an incomplete one and the reason for this is that the potential of which it is the actuality is incomplete. This makes it hard to grasp what change is. For it has to be assigned either to privation or to potentiality or to simple actuality.  But none of these seem to be possible.

$$201^b31$$

Eventually, Aristotle assigns it to a 'special kind of actuality'. That Aristotle struggled with this question is to understand that Newtonian mechanics is not really about change, it is more akin to geometry. It is motion tackled by geometrical means. And the same is true for it's classical completion, GR. However, the interpretational question that Aristotle found elusive reared it's ontological head in QM. The quantum wave describes change at the fundamental level, in a sense, it is our unit of change, but what is the ontological status of a quantum wave? Like Aristotle, we can certainly say it is not actual and only when it is 'complete' (to use Aristotle's language) or 'measured' (to use QM language) is it actual in the sense an observable yields up a measurable value.

One that doesn't seem to have been mentioned so far, from Lagrangian mechanics:

Forces are potential energy gradients. Wherever a change in the configuration of system would increase the potential energy in the system, there is force opposing that change.

The sum of $$-F\cdot d$$ over all the moving parts determines how much potential energy is increased by any small change, and motion against or with a force is in fact the mechanism by which energy is transformed from kinetic to potential form and vice-versa.

In short, potential energy wants out, and force is the expression of that.

I'm a bit out on a limb here but I think all the existing answers are missing the point, so I'm going to add my 2 cents.

The definition of a physical quantity is how you measure it.

Any definition in terms of mathematical or common language reference to other quantities is begging the question, "okay, but how do we measure those?" Force is mass times acceleration. Okay, acceleration is what an accelerometer measures, but mass is the ratio of acceleration to force, and we're right back where we started.

Force is what a calibrated scale measures.

A scale is a machine that measures the difflection of a known spring element and applies Hooke's law to relate force as a linear function of displacement.

Ultimately we could calibrate a scale using only times and distances, starting with the a priori definitions of $$c$$ and $$h$$ as basis constants, by looking at the work done on a beam of light in an interaction, as energy per quantum if light is a function whose only inputs are distance, time, and fundamental constants.

There may be other machines that will output the same value as a calibrated scale. If so, force could be equally well defined as what one of those machines measures.

I really don't think one can do better than to say a force is the measure of a specific kind of interaction. It is interaction, not "force", that is the most fundamental concept; and I don't think there's a better definition of this than its usual meaning: "what is happening amongst a number of things when they are having an influence on each other by their mutual presence". This is like your definition 2), but note that it is not a definition of force.

One can consider that the most basic object in classical mechanics of a single point-like object is its kinematic state:

$$(X, \mathbf{p})$$

where $$X$$ is the position, a point, and $$\mathbf{p}$$ is the momentum, a vector. To describe the system over time, we assign a state to it at each point in time. "Influence", then, shows up as a change therein, and we can translate "interaction" into agents of such change. We like to represent these interactions as having a summatory effect: if we let the state be denoted by $$\mathbf{с}$$, then

$$\frac{d\mathbf{с}}{dt} = \text{(sum of all 1-body interactions)} + \text{(sum of all 2-body interactions)} + \cdots$$

Force, then, is the measure of effect each interaction individually contributes to the change in the momentum:

$$\frac{d\mathbf{p}}{dt} = \text{(sum of all 1-body forces)} + \text{(sum of all 2-body forces)} + \cdots$$

where there is a 1:1 correspondence between interactions and forces.

When it comes to the "inherent behavior" of objects, though, there isn't really so much a way we can access it as that we have to invent a description language suitable for it. The above is one of many languages we can invent for such a purpose. (*)

But I also submit this does not make the question meaningless, for while those languages are invention, it is far from nonsense to ask what the meaning is that we have invented or created, or can invent or create, for a word or term therein that is suitable and appropriate to how it behaves. A language where nobody knows what the terms mean is quite a useless one, regardless of its fundamental ontological status. Although, of course, some theorists of language might argue that language is inherently empty of meaning, but if we wanna go down that rabbit hole then we are both a) really far from physics now and well into the realms of philosophy, especially non-analytic philosophy, and b) I am not sure if it's useful to actually trying to get one's head around the theory in question.

(*) Why choose a sum, for example? Actually, we don't have to. However we find empirically that, for some reason, things "factor more nicely" with sums; e.g. gravitation could be many different interactions together. Of course - maybe it really is. We don't know, and maybe we need to not be so dogmatic as to just avoid ever entertaining such ideas!

Force is the rate of change of momentum. The total momentum of an isolated system is constant, but its parts can mutially exchange momentum. A change in momentum of one part is then accompanied by an equal but opposite change of momentum of some other part. The rates of change are opposite, which constitutes Newton's third law.

Okay. One more. I think force is that, in a freely falling frame, what causes free particles to have velocities, in the mean, that change over time. Kinetic energies change. As do potential energies.

Free particles can collect in stable structures. Forces can balance. No change in kinetic energy will be seen although forces are at work.

What causes force? Charge. Charged particles want to be with each other or they don't.
Curvature of spacetime causes tidal forces that drives particles away from each other. The falling of a stone on Earth is caused not by the curvature of spacetime but by electric charges keeping up the surface of the Earth.

If there weren't charges in particles (only mass), all particles would have constant velocities, but tidal effects would cause relative accelerations. Which can be called forces. If no particles had charge all mass in the universe would collapse to black holes, though if they didn't have charge from the start, it remains to be seen if they could collect in lumps, and it could even be asked if they could have mass or exist at all.

I would argue that Newton's first law defines what is meant by a "force". Many mistakenly believe that Newton's first law is redundant and can be derived from Newton's second law $$F=ma$$ but this is wrong. If that were the case then why would there even be a "Newton's first law"?

Law 1. A body continues in its state of rest, or in uniform motion in a straight line, unless acted upon by a force.

What is a force? A force is something that disturbs a body's state of rest or its uniform motion in a straight line.

Edit: The mathematics of Newton's first law $$F=ma=0$$ can be derived from Newton's second law. But that would be to completely miss the point of Newton's first law.

Second Edit: The definition of Newton's first law above was taken from the top part of "Newton's laws of motion" wikipedia page. However further down the wording "net force" is used instead of "force" which is more accurate.

• "Something that disturb's a body state of uniform motion." Is it fair to say that this definition applies to absolutely any force in physics? Thank you. Commented Feb 28, 2022 at 21:30
• I would say yes this definition applies to all of the four forces; electromagnetic force, gravitional force, strong force and weak force. I am as @Bob D a bit unsure if it applies to strong and weak force but I would guess it does.
– ludz
Commented Feb 28, 2022 at 22:06
• @EthanDandelion No, this is obviously a definition of force that only works in classical mechanics -- as would be the case with any definition of any concept in classical mechanics.
– user87745
Commented Feb 28, 2022 at 23:04
• @BobD After sitting down and thinking about it for a couple of days I think this definition may actually be viable, if we use my previous definition: "Something that disturb's a body state of uniform motion." This is TECHNICALLY correct because if the body is already at rest, adding a singular force will disturb that state. However, my thinking now is that this definition simply isn't descriptive enough. To address this disturbance of uniform motion we effectively have to ignore all of the forces keeping it in equilibrium. Commented Mar 5, 2022 at 0:01
• @EthanDandelion If the object is sitting at rest next to a wall and we then add a singular force then it will not "disturb the state" since the wall will immediately counter this. It still needs to be net force.
– ludz
Commented Mar 5, 2022 at 9:56

A NET force is an interaction that causes mass to accelerate in the direction of the net force. Since there are only two known macroscopic fundamental forces, that interaction must necessarily be either electromagnetic or gravitational for forces applied to macroscopic objects.

• Hmm… Pauli exclusion causes deceleration upon contact, detectable as a net force, but is not one of those two fundamental interactions. Commented Mar 1, 2022 at 0:23

I guess one way to think about it in a generalized, abstract way, is in the style of abstract mathematical definitions (e.g. a topological space is any object of the form [set + some subsets] that satisfies such and such axioms).

So, in that spirit (but without attempting that level of rigor), perhaps one could go about it this way.
Within the context of Newtonian dynamics, you could say that there are the following building blocks*:

1. A notion/concept that there can be these directed influences of different magnitudes between objects that act either on contact or over distance, that are capable of changing the state of motion of objects they act upon, and can be represented as vectors**. Qualitatively, these influences "push" or "pull". These are then termed "forces".

$$\hspace{9pt}\large \&$$

1. The idea that, given a description (model) that explains the dynamics of a system, some elements of the model (vectors, vector-valued functions, ...) can be designated as forces iff these model elements exhibit roles and behaviors expressed by Newton's laws (law of inertia, $$\textbf F=m\textbf a$$ or equivalents, action-reaction).

Inline footnotes:
* These are probably incomplete (additional assumptions could potentially be included, like flat spacetime).

** Vector here being an abstract mathematical object (so, distinct from how one chooses to represent it (e.g., from a particular coordinate representation).

Then, when you are able to successfully map this framework onto an actual physical situation (and you may not be able to) and onto the accompanying contextualized mathematical description, you call "force" whatever ends up being mapped to the concept of the force as defined above.

Further, if within a wider theoretical conceptualization there's an explanation for the origin of the force, then it's considered a real force (an actual physical thing), if there isn't, then it's a fictitious force. (And then perhaps you could build on top of that to define inertial frames, etc. I know this seems backwards, but, if you start with inertial frames, then you have to bolt on fictitious forces, and explain the sense in which these are "forces", why use the term, and so on. I think this works, but I could be missing something.)

In other words, if it's a vector-like influence, and if it is behaving in accordance to Newton's laws, it can be called a force in some sense compatible with $$1\ \&\ 2$$ (if it looks like a duck, and quacks like a duck, ...) — unless there exists within a larger theoretical context some overriding reason that modifies the meaning, or alters the fundamental description of the system (e.g. when people invoke GR and say that gravity is not really a force in the above sense, but that you can treat it as such under certain circumstances).

Also, in a larger theoretical context this conceptualization can be related (perhaps with caveats/constraints) to other qualitatively different ideas (e.g. negative gradient of the potential), equivalences can be established between different theoretical descriptions, etc.

And then there are uses of the term outside of this context, where the meaning may be more loose, and Newtonian mechanics might not even apply (e.g. strong interaction being called strong force).

An ontological answer to round up all the other attempts:

What is the fundamental definition of force?

Since you know of $$F=ma$$ and do not accept that as your answer, I guess you want to know what force "really" is. Objectively, in the universe, without any human concepts or "theories".

Herein lies the crux. This kind of answer cannot be answered. This is the age old problem of Plato's cave. We do not know the answer to questions of that kind for any physical phenomenon.

We do not know what an electron actually is. We do not know what a photon is. We don't know what electromagnetic fields really are. All we know about anything is our models, which describe some aspects of the behaviour of physical objects in relationship to other objects at varying levels of detail and correctness.

Yes, we have become quite refined, and our knowledge is substantial. We've split the atom and watched the big bang, but we still are not a single bit of knowledge closer to knowing what all that stuff actually is. It would still be quite possible that I am just a brain in suspension, with all my nerve cells connected to some computer feeding me totally wrong information. Or I could still be a simulation in a computer, just like you and everyone else, in an actual universe which behaves totally different than what we are fed.

(Seeing all the other answers, my own best guess at a helpful answer would be to simple re-iterate Newton's First and Second Law - for all intents and purposes, everything that makes objects move is a force and vice versa, by definition - fully accepting that this is only useful for the electromagnetic force and maybe not so much for the other three fundamental forces).

• Quoting the answer physics.stackexchange.com/a/697000/226902 , "In Newtonian mechanics, a force is a mathematical vector we prescribe onto a model of a physical system by declaring a force law.". This is "really" what a force is, because this is exactly the description of the concept of "force" that is used ever yday by "real" humans to make calculations on "real" paper or running simulations on "real" computers. Nature would probably work the way it does with no regard for humans inventing and using a formal "force" concept. Commented Mar 2, 2022 at 10:51
• True about "reality", but the question also asks about a definition. That is part of a human model, not something separate from it. Hence, we can ask whether the term force is given definition. Commented Apr 26, 2022 at 2:43

A comprehensive and exhaustive definition of force is yet to evolve through discussions, studies, in depth analysis etc. F=ma helps to calculate the magnitude of force in cases where motion is involved.

In the gravitational field for example, every cubic centimeter space in the gravitational field is filled with 'Force'. The moment an object with mass and volume is brought into that space, it experiences force in the form of 'gravitational pull'. It will experience force irrespective of spot 'A' or spot 'B' in the gravitational field. The object is only a medium for the gravitational force to act. Irrespective of whether object is there or not, the 'causal gravitation pull' is present through out the gravitational field through out the time (Is it a wave?)

In an IC engine, when fuel burns inside the cylinder, high pressure is developed which exerts the force/pressure on the piston which in turn causes the shaft to rotate.Here, force is derived by burning of fuel, and that force is obtained during the combustion stroke only and not through out the time.This is a case force being developed by spending energy over a period of time, as opposed to 'gravitational force' which is present all the time without burning any fuel.

In the case of magnetism, both attraction and repulsion forces are there. Only magnetic materials experience magnetic force. Between magnets, there is attractive and repulsive force. To explain away all these, we need much more insight.

This discussion is inconclusive. We can rest assured that a comprehensive out look will emerge as the discussions, exchange of ideas proceed.

As you probably know, one of the observations about the universe is that there exist frames of references where particles move in straight lines when they seem to be far away in isolation from all the other particles.

But in general, the goal of physics is to predict the trajectories of interacting particles that are not in isolation (non-straight lines in general).

Then it follows that the equations of physics should be able to tell us the acceleration of a particle, if we provide the current state of the system as input (positions and velocities of all the particles in the system).

Acceleration is the quantity we care about in the end, as the goal of physics is to predict trajectories.

Now, how do we actually make inferences about how acceleration behaves? Obviously: by making observations about acceleration.

Observation 1: In an isolated system of two particles (say, billiard balls), their accelerations at any point of time during the collision is observed to have the property:

$$-\frac{a_1(t)}{a_2(t)}=c$$

The minus sign indicates the opposite directions of $$a_1$$ and $$a_2$$. $$c$$ is always the same positive constant. This observation allows you to say "Particle 2 is $$c$$ times as massive as Particle 1". We're defining relative mass right now. In any interaction involving the two particles, particle 1 "suffers" $$c$$ times as much change in velocity as particle 2, in the same amount of time.

We've defined mass as the constant measuring measuring resistance to change in the state of motion. We can further choose a reference mass, call it 1 unit, and relative to it, ascribe mass values ($$m_i$$) to all the particles of the universe. Then the equation would become:

$$-\frac{a_2}{a_1}=\frac{m_1}{m_2}$$

or $$m_1a_1+m_2a_2=0$$

or $$m_1v_1+m_2v_2=constant$$

Moving on, so far we've only observed a relationship between two accelerations $$a_1$$ and $$a_2$$ and that happen in pair. There's still no formula to measure either $$a_1$$ or $$a_2$$, given the initial state of the particles (positions, velocities).

Enter gravitation. The acceleration formula that has been observed to explain the motion of planets is $$\frac{Gm}{r^2}$$, $$m$$ being the mass of the sun and $$r$$ being the distance between Sun and Earth.

For the first time, we actually are able to calculate the acceleration in a system of two interacting bodies.

But, hold on, this acceleration formula coupled with the previous observation is enough to calculate the trajectory of the system. Then, who cares about Force?? Why introduce a middle man called Force??

I mean..you could introduce a quantity with the formula $$F_1=m_1a_1=\frac{Gm_1m_2}{r^2}$$. Then the calculation of acceleration ($$a_1=\frac{F_1}{m_1})$$ would go one step down the road for no reason. Why introduce Force when acceleration itself has a much simpler formula?

For one, you can't deny Force has a convenient property. From the first observation, $$m_1a_1=-m_2a_2$$. Unlike acceleration, this guy occurs in equal an opposite pairs.

For another, gravity isnt't the only interaction that we've observed:

Enter electromagnetism: Now we're in a realm where the observed laws are best described in terms of the quantity $$ma$$. The observed laws are $$F=k\frac{q_1q_2}{r^2}$$ and $$F=qvB$$. $$m$$ no longer simply cancels out in the division. In fact, it sticks around and acceleration ends up having an uglier formula.

This, combined with the "equal and opposite property", now makes Force the natural quantity to work with. Acceleration becomes the derived quantity by $$a=\frac{F}{m}$$

Another reason Force is a convenient quantity: Work energy theorem

With just gravity, you could simply write down the work energy theorem as:

$$\frac{1}{2}(v_2^2-v_1^2)=\int \frac{GM}{x^2}dx$$

No force involved in the above formula. You could forcefully involve it by multiplying both sides by $$m$$.

This again changes in electromagnetism:

$$\frac{1}{2}m(v_2^2-v_1^2)=\int k \frac{q_1q_2}{x^2}dx$$

This is a fundamental equation in which the RHS is naturally an integral of Force. You can no longer cancel out the mass