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Newton's second law

$$\mathbf{F} = m\mathbf{a}$$

where $\mathbf{F}$ is the force, $m$ the mass, and $\mathbf{a}$ the acceleration, seems at first blush to be a simple tautology, since $\mathbf{F}$ and $m$ are not defined anywhere else in the formalism. Of course we can use the more general formulation

$$\frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt}$$

with $\mathbf{p}$ the linear momentum and $\mathbf{v}$ the velocity, but given the usual elementary definition of linear momentum $$\mathbf{p} \equiv m\mathbf{v}$$

this seems hardly an improvement.

Have I missed something? Perhaps in the Lagrangian or Hamiltonian formulation one can use Noether's theorem etc. to give a less trivial definition of $\mathbf{p}$? Or is there something even more elementary than that which I've missed?

Now, the law is clearly not tautologous in concert with some kind of force law, such as Coulomb's law or the law of universal gravitation. In that case we know what force means independently of mechanics. I'm just wondering if mechanics can be viewed as logically self-contained without recourse to such an addition.

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    $\begingroup$ @Vishwaas that is a common misconception, the equation only works if you assume constant mass $\endgroup$
    – user83548
    Commented Nov 19, 2015 at 17:00
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/70186/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Nov 19, 2015 at 17:14
  • $\begingroup$ This question is actually about the meaning of “force” in Newton’s laws. In practice, one finds the form of the force in the second law by observing the motion of bodies under such a force (i.e., the change in their momentum); e.g., looking at the orbits of heavenly bodies, and deducing the $\frac{1}{r^2}$-dependence of the gravitational force. $\endgroup$
    – AlQuemist
    Commented Nov 19, 2015 at 17:55
  • $\begingroup$ As you may have heard from general relativity, the "force of gravity" is not really a force in the conventional sense; in fact it is again chosen so as to fit with the dynamical observations, Kepler's laws in particular. One may postulate a reference force formula like the weight of an object or Hooke's law for instance, but then the meaning of the second law becomes dependent on this postulate and its violation could mean either that the second law is wrong or that the constitutive law is wrong: how do you choose which is the case? $\endgroup$
    – gatsu
    Commented Nov 19, 2015 at 18:35
  • $\begingroup$ Regarding the title question, at least: I am reminded of Wittgenstein's observation that all propositions are tautologies. Newton's Laws are declarations of equality. They are saying, "this is true, always". They are defining tautologies, by saying, "look at this thing on the left side. If you write it a little differently it's the thing on the right". $\endgroup$
    – commando
    Commented Nov 19, 2015 at 19:14

4 Answers 4

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Newton's first law states that there exist very special reference frames (that we are going to call inertial henceforth) where any point particle not subject to external forces (interactions) moves in straight lines, i. e. the equation of motion is $\dot{\textbf{p}}(t)=0$.

Newton's second law states that, in the above mentioned reference frames (and only in the above), whenever a point particle is subject to external forces $\textbf{F}(\textbf{r}, \textbf{p})$ the equation of motion modifies to become $$ \dot{\textbf{p}}(t)=\textbf{F}\left(\textbf{r}(t), \textbf{p}(t)\right). $$ Namely, force and variation of momentum are by themselves two different things that happen to be equal in those special reference frames, for point particles subject to external interactions.

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  • $\begingroup$ As far as I am aware Newton never mentioned the linear momentum. He did say however that the change of the state of motion was proportional and directed in the direction of the force impressed. From that however, I am not sure how one can infer whether the concept of force is a primitive one or derives from the acceleration. $\endgroup$
    – gatsu
    Commented Nov 19, 2015 at 17:37
  • $\begingroup$ @gatsu The key point is that force and momentum (or acceleration, or $mv$, or whatever other state of motion variable) are, to start with, two different things that have their own independent definitions. Then, they "accidentally" happen to be the same, for the point particle, is some (inertial) reference frames. $\endgroup$
    – gented
    Commented Nov 20, 2015 at 0:05
  • $\begingroup$ I perfectly appreciate that it is a possible interpretation but I fail to see how it is deducible that it is an equality and not a definition. For one thing, one needs a mean to measure independently the quantities on each side of the equality sign. This is only possible provided one postulates a constitutive law for forces upon which the equality will be pending. $\endgroup$
    – gatsu
    Commented Nov 20, 2015 at 11:55
  • $\begingroup$ @gatsu It is perfectly possible to measure the two quantities independently: the standard example is to measure the force using a torque, or a spring, or any other mechanically balanced scale tool independent of the accelerations. This ensures that the force can be defined and measured independently. $\endgroup$
    – gented
    Commented Nov 20, 2015 at 13:00
  • $\begingroup$ yes, yes it is as standard experiment that relies on the concept of forces in statics. Note that I am not disputing the fact that force can be measured independently, I am just saying that if that's the case, then it comes from a somewhat ad hoc constitutive law that has to assign for the first time a number to a single force; and nothing is obvious about such a task. To take weight as an example, nothing obliges us to assume that weight has to be linear in the amount of matter we put on a balanced scale; the only constraint such an instrument requires is balance of forces. $\endgroup$
    – gatsu
    Commented Nov 20, 2015 at 21:53
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Suppose a force $F$, acting on object $A$, produces acceleration $a_A$, and the same force $F$, acting on object $B$, produces acceleration $a_B$. Suppose a different force $G$ produces accelerations $a_A'$ and $a_B'$. Then Newton's law implies $a_A/a_B=a_A'/a_B'$, which is not tautological.

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  • $\begingroup$ The above only holds if you assume the masses to be constant (otherwise you cannot just divide by them, as the derivatives on the RHS bring up two different contributions). $\endgroup$
    – gented
    Commented Nov 19, 2015 at 17:18
  • $\begingroup$ @GennaroTedesco: The question was about Newton's second law, which does in fact assume the masses to be constant. $\endgroup$
    – WillO
    Commented Nov 19, 2015 at 17:22
  • $\begingroup$ There is no assumption on the masses to be constant in Newton's law (in fact they are formulated in terms of the momentum and not in terms of $ma$). $\endgroup$
    – gented
    Commented Nov 19, 2015 at 17:25
  • $\begingroup$ @WillO While this is a good answer, I undid the acceptance due to the caveat that it only applies under constant mass. Is there a simple generalization to more complicated mass relationships? $\endgroup$
    – AGML
    Commented Nov 19, 2015 at 17:28
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    $\begingroup$ AGML and @GennaroTedesco: If mass is variable, then $F=ma$ is not true. The question was about the meaning of the law $F=ma$, which is a law only when mass is constant, so surely any answer will apply only when mass is constant. $\endgroup$
    – WillO
    Commented Nov 19, 2015 at 17:37
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I believe there is no definite answer to this question and it has puzzled physicists for a long time. In fact, it has puzzled them so much that after the grand work of Newton's, many people still sought different formulations.

The Cartesian school for instance thought that "only pre-existing motion can trigger a change of motion" akin to what people witness in collision experiments. From these types of principles arose rules such as the conservation of momentum and energy which at no moment need the concept of force; their role in fact is simply to constrain the type of dynamics undergone by physical objects nothing else. In this case force may be understood as the name for change of motion and it derives from dynamics. While they are not enough on their own and need to be supplemented by some other principle, it is still gives some insight on how one can view the concept of force as deriving from dynamics.

Euler and Daniel Bernoulli on the other hand seemed to think of force as a primitive concept about which one made propositions (including the 2nd law) and it is likely the case that most thinkers who view(ed) the paradigmatic example of force in static problems (rather than dynamical ones) think of force as a primitive concept.

To add to the list, are some variational formulations of mechanics: Gauss's least constraint formalism seems to be a way of enforcing Newton's second law by considering that force and change of momentum can be different things, however, the least action principle states something quite different that does not involve any force concept per se.

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F and m are not defined anywhere else in the formalism

The force F is generic. In each particular case it is replaced by a force that is defined. For example $F = kx$ or $F = KMm/r^2$.

$m = Weight/g$ and the weight is measurable with a balance.

So everything is defined.

What I know is that the Aristotelian view, before Newton, was that:

$F=kv$

where $k$ was an universal constant and $v$ the speed.

The second law of dynamics $\mathbf{F} = m\mathbf{a}$ is not evident at all.

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