Past exam of the philosophy of space-time and philosophy of quantum mechanics paper of Parts A and B of the physics course of the University of Oxford / 2019 / 1

There is no unconditional logical dependence of Newton's first law on Newton's third law. There is, however, a dependence if one applies the first to an extended, isolated composite and interprets its motion as the motion of its centre of mass. The difference is between the behaviour of an elementary body with zero total force and a theorem about cancellation of internal forces. Isaac Newton's own first-law discussion includes projectiles and spinning bodies, so the composite-body issue is physically significant.^[ref]

For a constant positive mass in an inertial frame, Newton's second law gives $m\ddot{\mathbf{x}}=\mathbf{F}$. Setting the total force to zero yields uniform rectilinear motion; it uses no action–reaction assumption. The first law additionally identifies a common class of inertial frames in which such a statement applies. One cannot establish that class by arbitrarily straightening a single trajectory, and one cannot test the claim by defining every observed acceleration to be an unspecified force.

A concrete countermodel separates these issues. Suppose two bodies obey the second-law equations with internal forces ${\mathbf{F}}_{12}=\mathbf{f}$ and ${\mathbf{F}}_{21}=-2\mathbf{f}$, with nonzero constant $\mathbf{f}$. Every elementary body with zero total force still has constant velocity, but the two-body momentum satisfiesThus the elementary-body first law is compatible with failure of the third. Calling the pair isolated does not make its internal total force zero: that is precisely what has not been assumed. This countermodel establishes logical independence, not a proposal for a successful fundamental interaction.

Now strengthen the first-law claim: every isolated composite has an inertially moving centre of mass whenever the total external force vanishes. With $M={\sum }_i{m}_i$, Newton's second law givesFor an isolated pair, the strengthened claim requires ${\mathbf{F}}_{12}+{\mathbf{F}}_{21}=0$. This is the weak, equal-and-opposite form of Newton's third law. It does not by itself require collinearity with the bodies' separation; the central-force condition is stronger and bears on conservation of angular momentum. For many bodies, cancellation of the sum does not alone imply pairwise cancellation without assumptions about the decomposition into interactions.

The right conclusion is therefore conditional. A first law for individual force-free bodies does not presuppose the third law. A first law for all isolated material composites, combined with the usual centre of mass and Newton's second law, imports an internal-momentum-balance condition. In theories with dynamical fields, material bodies can exchange momentum with those fields; conservation of momentum then concerns the combined system and need not imply instantaneous pairwise action–reaction between particles.

The familiar result follows if the force laws have Galilean invariance; the three abstract Newton's laws of motion do not guarantee that invariance. Isaac Newton's Corollary V correctly identifies a central feature of ordinary Newtonian mechanics, but its general physical scope requires assumptions about interactions as well as about the relation between force and acceleration.^[ref]

Take a constant velocity $\mathbf{U}$ and a Galilean transformationThen ${\ddot{\mathbf{x}}}_i^{\prime }={\ddot{\mathbf{x}}}_i$, ${\mathbf{x}}_i^{\prime }-{\mathbf{x}}_j^{\prime }={\mathbf{x}}_i-{\mathbf{x}}_j$, and ${\dot{\mathbf{x}}}_i^{\prime }-{\dot{\mathbf{x}}}_j^{\prime }={\dot{\mathbf{x}}}_i-{\dot{\mathbf{x}}}_j$. Any interaction depending only on these relative quantities, with unchanged parameters, supplies the same forces for the transformed initial data. Newton's second law consequently gives the same relative accelerations; uniqueness of the resulting initial value problem supplies the same relative histories. An inverse-square central interaction is a standard example. Newton's third law helps establish the common centre of mass result, but the boost argument principally needs invariant interactions and the second law.

Here is a counterexample to deduction from the three laws alone. Give two equal masses a preferred inertial frame and write $\mathbf{r}={\mathbf{x}}_1-{\mathbf{x}}_2$, $\mathbf{V}=({\dot{\mathbf{x}}}_1+{\dot{\mathbf{x}}}_2)/2$. For fixed $k>0$ and speed scale $u_{\ast }>0$, specifyThe forces are equal, opposite and central, the centre of mass is inertial, and each body obeys $m{\ddot{\mathbf{x}}}_i={\mathbf{F}}_i$. An elementary body with no force is inertial. Nevertheless,Boosting an isolated pair from $\mathbf{V}=0$ to $\mathbf{V}=\mathbf{U}$ changes its internal oscillation frequency. All three motion laws hold, but the proposed universal boost equivalence fails. The extra preferred structure in the force law is what makes the failure possible.

A passive coordinate transformation of this same model remains possible: transform the preferred frame's description too. That mathematical freedom does not establish an active dynamical symmetry which changes the physical pair's velocity relative to the preferred structure while holding the laws' background data fixed. Confusing those operations would hide the counterexample.

Finally, a change to rotating coordinates introduces centrifugal forces and Coriolis forces; their dependence on position or velocity can change internal experiments. Constant translational velocity has no analogous effect when the interactions have Galilean invariance. Corollary V expresses that physical symmetry, with the interaction assumption made explicit, rather than a consequence of action–reaction alone.