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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.

Qmechanic
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AGML
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4 Answers4

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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.

gented
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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.

WillO
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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.

gatsu
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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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