Newton’s Third Law: “Action Equals Reaction”

​In Chapter 10-1 of The Feynman Lectures on Physics, Feynman situates Newton’s Third Law within a broader methodological discussion concerning the limits of analytical and numerical approaches in classical mechanics. While Newton’s Second Law provides, in principle, a complete prescription for determining motion from known forces, Feynman emphasises that practical and conceptual considerations motivate a deeper examination of the laws themselves. Analytical solutions, where they exist, reveal general structures of motion - parabolic trajectories under uniform acceleration, harmonic oscillations described by trigonometric functions, and elliptical planetary orbits - that numerical methods, though powerful, do not naturally expose. These solutions contribute not merely computational efficiency but conceptual understanding.



Feynman then contrasts such analytically tractable systems with those that resist closed-form solutions. Even modest increases in complexity, such as deviations from linear restoring forces or the addition of a third gravitating body, render purely analytical treatment impractical. The historical difficulty of the three-body problem serves as an example of the inherent limitations of mathematical analysis. Although modern numerical methods can now handle such systems routinely, Feynman stresses that there exist classes of problems - those involving enormous numbers of interacting particles, such as gases, solids, or stellar systems - for which neither analytical nor direct numerical methods are feasible. In these contexts, physics must rely on general principles that do not depend on detailed microscopic trajectories.


It is within this framework that Newton’s Third Law acquires its significance. When detailed motion cannot be followed, the focus shifts to global properties of systems, particularly conserved quantities. Feynman identifies conservation of momentum as one such principle, directly rooted in the Third Law. The law states that when two particles interact, the force exerted by the first on the second is equal in magnitude and opposite in direction to the force exerted by the second on the first, with both forces acting along the same line. Crucially, these forces act on different bodies and therefore do not cancel in the dynamical equations of either particle individually.


The scope of the Third Law is explicitly pairwise. In systems containing more than two particles, the total forces acting on individual bodies need not be equal or opposite. However, by resolving the forces into components associated with each interacting pair, one finds that each mutual interaction satisfies the Third Law independently. This pairwise symmetry ensures that, when considering a closed system, internal forces cancel in the aggregate. As a consequence, changes in total momentum can arise only from external forces, leading directly to the conservation of momentum for isolated systems.


Feynman also notes that Newton’s Third Law represents nearly the entirety of Newton’s insight into the general nature of forces beyond gravitation. While Newton possessed no detailed knowledge of atomic or contact forces, he identified a universal structural feature of interactions: their reciprocal character. Although later developments in physics have revealed limitations to the Third Law in its classical form - particularly in electromagnetic systems where momentum may be stored in fields - the law remains an extraordinarily accurate approximation in classical mechanics. More fundamentally, conservation of momentum is now understood as a consequence of the translational symmetry of space, with Newton’s Third Law emerging as a classical manifestation of this deeper principle.


Thus, Chapter 10-1 establishes Newton’s Third Law not as a redundant addendum to the Second Law, but as a foundational principle enabling the transition from detailed dynamical descriptions to general conservation laws. It provides the conceptual basis for analysing collisions, many-body systems, and collective phenomena, and marks a shift from force-centred calculation to symmetry-based reasoning in classical physics.

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