= Constraint power transfer
\b[Constraint power transfer] is redistribution of <energy> between parts of a mechanically connected system through <constraint forces>. An ideal connection may exert nonzero <power> on each body while exerting zero total power on the combination. Power on a body is the actual force–velocity scalar product; the zero-<virtual work> property of an <ideal mechanical constraint> concerns the sum over admissible virtual displacements.<Advanced Classical Mechanics by Iain W. Stewart>[§1.4, equations 1.62 and 1.64–1.66; constrained-system virtual work]{ref}
An original <ideal rigid link> calculation makes the cancellation explicit. Let two endpoint positions be $\mathbf r_A$ and $\mathbf r_B$, joined by a massless link of fixed length $\ell$. Set
$$
\mathbf n=\frac{\mathbf r_B-\mathbf r_A}{\ell},\qquad
\mathbf n\cdot(\mathbf v_B-\mathbf v_A)=0.
$$
For axial tension $T$, the link exerts $+T\mathbf n$ on A and $-T\mathbf n$ on B. The powers supplied to the endpoints are
$$
P_A=T\mathbf n\cdot\mathbf v_A,\qquad
P_B=-T\mathbf n\cdot\mathbf v_B,\qquad P_A+P_B=0.
$$
The endpoint velocities need not be identical: the link can rotate. Only their components along the link must agree.
For stipulated $T=12\,\mathrm N$ and common axial endpoint speed $0.40\,\mathrm{m/s}$, the link supplies $+4.8\,\mathrm W$ to A and $-4.8\,\mathrm W$ to B. It transfers energy without supplying net energy to their combined system. A load cell measuring axial force and tracked endpoint positions measuring velocities would permit these two powers to be compared. External forces determine how the endpoints subsequently accelerate.
Equal and opposite forces alone do not prove power cancellation. If the connector stretches, $\dot\ell=\mathbf n\cdot(\mathbf v_B-\mathbf v_A)$ and the same axial-force calculation gives
$$
P_A+P_B=-T\dot\ell.
$$
An elastic connector can store the corresponding <elastic energy>; a dissipative connector can convert mechanical energy into heat. Likewise, an externally prescribed moving support can deliver nonzero net <constraint power> to a subsystem. The location of the system boundary determines whether a measured exchange is internal transfer or externally supplied work.
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