= Newton's third law
{c}
{title2=$\mathbf F_{A\mathrm{\ on\ }B}=-\mathbf F_{B\mathrm{\ on\ }A}$}
In the pair-interaction model of <Newtonian mechanics>, two bodies exert equal and opposite <forces> on each other:
$$
\mathbf F_{A\mathrm{\ on\ }B}=-\mathbf F_{B\mathrm{\ on\ }A}.
$$
Both <vectors> are expressed in the same <inertial frame> at the same <time>. The subscripts name the agent and the recipient: $A$ acts on $B$, and $B$ acts on $A$. These are <forces> on different bodies, so they cannot cancel in the net <force> on either body. <Newton's second law> still uses all the <forces> on the particular body whose <acceleration> is sought. Identifying this <third-law interaction pair> does not require the bodies to be isolated: additional <external forces> can act at the same <time>.
The distinction makes <Newton's third law> useful both for drawing a <free-body diagram> and for deriving <momentum conservation>. A body-by-body calculation retains one member of each <third-law interaction pair>; a calculation for a collection containing both bodies cancels that pair in the total <momentum> balance. Equal interaction <forces> consequently coexist with nonzero, generally unequal <accelerations>.
<Isaac Newton> stated the equality and opposition of mutual actions in Law III of his 1687 \i[Philosophiae Naturalis Principia Mathematica]. The explanation distinguishes equal opposite changes of <momentum> from changes of <velocity>, which are inversely proportional to the bodies' <masses> when other influences are absent. Corollary III then explains why mutual interactions preserve the signed total <momentum>. The equations here are modern notation for that distinction. <Philosophiae Naturalis Principia Mathematica by Isaac Newton/Axiomata sive leges motus>[Isaac Newton, Philosophiae Naturalis Principia Mathematica (1687), Axiomata sive leges motus, printed p. 13, Lex III and its explanation; pp. 16–17, Corollarium III]{ref}
The instantaneous body-pair equation has a scope: use it for the mechanical interactions assumed by this <Newtonian mechanics> model. In <electromagnetism>, bodies can exchange <momentum> with an <electromagnetic field>; their instantaneous mutual <forces> need not sum to zero. <Momentum conservation> then accounts for <electromagnetic field momentum> as well as matter, and for <momentum> crossing the chosen boundary. This complication occurs in classical <electromagnetism>, without invoking <quantum mechanics>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others/Chapter 27: Field Energy and Field Momentum>[The Feynman Lectures on Physics, Volume II, §27–6, equation (27.21) and final two examples: momentum of matter, field momentum, and failure of particle-only action and reaction]{ref}
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