dRic2
Gold Member
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 80
 Problem Statement

Let us assume that the work function of an assembly of free particles, subject to mutual forces between these particles, depends only on the relative coordinates
$$\xi_{ik} = x_i  x_k$$
$$\eta_{ik} = y_i  y_k$$
$$\zeta_{ik} = z_i  z_k$$
of any given particle ##P_i## and ##P_k##:
$$U = U(\xi_{ik}, \eta_{ik}, \zeta_{ik})$$
Let the coordinate ##x_i## be varied by ##\delta x_i##, thus obtaining the xcomponent of the force acting at ##P_i##. Show that the quantity:
$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}}$$
can be interpreted as the xcomponent of the force on ##P_i## due to ##P_k##.
 Relevant Equations
 .
I tried to apply the chain rule
$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} \frac {\partial x_i}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} $$
and I got the force xcomponent of the force acting on ##P_i## I guess.
but I do not know what to conclude from this...
$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} \frac {\partial x_i}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} $$
and I got the force xcomponent of the force acting on ##P_i## I guess.
but I do not know what to conclude from this...