- #1

dRic2

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- Homework 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 x-component of the force acting at ##P_i##. Show that the quantity:

$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}}$$

can be interpreted as the x-component 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 x-component 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 x-component of the force acting on ##P_i## I guess.

but I do not know what to conclude from this...