PHY.4:5.2 - Add a physical direction

Now the body is oriented or the environment has an aligned surface. Include its unit direction n in the input. Under a rotation of the complete situation, both w and n change, so the condition becomes:

F(Qw,Qn)=QF(w,n).

A rotation fixing w can now change n. The earlier argument that fixed all inputs while rotating a perpendicular output component is no longer available.

For example, one admissible linear resistive family is:

F(w,n)=-alpha*w-beta*(n dot w)*n.

For unit n, its power is -alpha*|w|^2-beta*(n dot w)^2. Resolving w into parts parallel and perpendicular to n shows passivity for every w when alpha>=0 and alpha+beta>=0. This is a possible family, not an exhaustive determination of the changed law.

With consistent units, take alpha=1, beta=3, w=(1,1,0) and n=(1,0,0). The force is (-4,-1,0) and power is -5. The force resists motion while pointing in a direction different from -w. This possibility is excluded by the earlier isotropic account and allowed by the additional physical input.

If n is fixed in the laboratory while only the body motion changes, retain that fixed n in the experiment’s account. Rotating the coordinate system changes the components of both quantities; it supplies no premise that removes the material or environmental direction.