[Physics] the fundamental definition of force

definitionforcesnewtonian-mechanics

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.

Best Answer

(Look at the section Some Further Clarification for a bit of meta-commentary on what we are trying to do when we are defining something. I think it has some important information.)

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).

This is 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 is never directly measured. Consider the example below.

Example 1. 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 declare the force law 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 manually, because this law was very successful for Newton to make astronomical predictions.

Another example is given below.

Example 2. 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.

One important point to understand is that neither Newton's first nor second law are used to define what is a force. It's the force law specific to the situation that defines the force, and then Newton's laws relate it to motion.

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. As far as we can tell the fundamental forces can be written in terms of fields, which are yet another slew of mathematical gadgets that we invoke. To define a field, we ascribe a vector (or tensor, etc) to every point in spacetime. The most well-known examples are the electric and magnetic fields.

Given a system with electric field $\vec{E} = \vec{E}(x, y, z, t)$ and magnetic field $\vec{B} = \vec{B}(x, y, z, t)$, the Lorentz force law states that the force on a particle of electric charge $q$ and velocity $\vec{v}$ is $$ \vec{F} = q\vec{E} + q\vec{v}\times\vec{B}. $$

Non-relativistic gravitation can also be put into a field-theoretic form described here. The force law for that is $\vec{F} = m\vec{g}$ where $\vec{g}$ is the "gravitational field" and $m$ is the "gravitational charge" in analogy with $\vec{F} = q\vec{E}$ for electric fields.

Some Further Clarification

I thought about this question some more, and I realized there are a few more points that need to be mentioned.

A lot of the other answers to this questions either rely on vague intuition or they define force in terms of other things and inevitably it shifts the burden on asking what those other things are (e.g. you can say force is a change in momentum per time, but then it leaves open the question of what is momentum). I think I can give an account for why this is the case.

Let me give a related example. What are lines and points in Euclidean geometry? For a long time, lines and points were considered primitive notions that don't have any explicit definition. They were primitive things that were characterized by axioms of Euclidean geometry (the axioms told us how we could treat these concepts but there was no explicit definition in the form of "a line is blah-blah-blah" or "a point is such-and-such"). However, around the 19th and 20th century, set theory began to be developed and people made a reformulation of geometry in terms of real analysis, which was itself founded on set theory. In this new formulation, the notion of a set was the primitive (not explicitly defined) notion, and everything else was defined in terms of sets. In particular, points and lines now had concrete definitions: a point on the plane is an ordered pair of real numbers $(x, y)$ and a line was a set of points $(x, y)$ such that $ax+by = c$ for some real constants $a, b, c$. Now lines and points could be explicitly defined in terms of other things.

Now to define force, we have two options:

  1. Option 1 is accept the notion of a force as a primitive concept with no explicit definition, and build axioms around how you want to characterize it.
  2. Option 2 is to start in a different theory (that has its own various primitive notions) and give an explicit definition of force in terms of the elements of that theory.

I think you can see pretty clearly how these options map on to the scenario involving points and lines in Euclidean geometry. Both options are perfectly tenable.

If we start with Newtonian mechanics, then mathematically speaking force is going to have to be a primitive notion. If we start with some other formalism like 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: $\vec{F} = -\nabla U$.

The above options are the only two ways you can define anything rigorously, and force just happens to be a primitive concept in Newtonian mechanics, because it starts with force.

Although force itself is primitive, it is supposed to be the mathematical concretization of the intuitive (but vague) notion of pushes and pulls (and more generally influences between bodies). The desired characterization that justifies force as the concretization of the notion of pushes and pulls is done through the axioms of Newtonian mechanics. You need to actually do and solve problems with Newtonian mechanics to understand exactly what this means.

Regarding Newton's Laws of Motion

As I've said, what exactly is the force in a given scenario is specified by the relevant force law. If you come across a new scenario that no one else has analyzed, you will have to guess the force law and empirically test whether or not your guess leads to correct predictions.

Of course, like I've said before, the force law can come from other theories such as electromagnetism where force is defined by the electric and magnetic fields.

Newton's first and second laws are not definitions of force so much as they are axiomatic characterizations of force. 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 (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). What the law truly means is that any time we have a force without an opposite force, the system we are analyzing is not truly a closed/isolated system.

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