if the external forces and the internal forces corresponding to an ensemble of particles are conservative. then referring to the potential that gives rise to the interaction force as Vij .. we consider that it is a function of the displacement vector |ri-rj| in order to follow the strong law of action and rtn .
how this assumption makes the interaction force a central force? ie it follows the strong law of action and rtn.
secondly.. what do you mean by grad(Vij)? and why is it equal to f*(ri-rj) where f is a scaler.. (1)
by (grad)i(Vij) we mean derivative wrt to ri.. but what exactly do we mean only by grad.. i know it will be wrt to r.. but how do we arrive at (1)
many particle system..
Re: many particle system..
First, grad of V cannot be scalar function.
Second, if V is central ( that is, ri-rj dependent) then grad of V is force(negative, more precisely) which is central too, and this should be pretty straightforward if you work in spherical coordinates (that is, write the gradient in spherical coordinates and you will obtain only r hat term--> central force)
Second, if V is central ( that is, ri-rj dependent) then grad of V is force(negative, more precisely) which is central too, and this should be pretty straightforward if you work in spherical coordinates (that is, write the gradient in spherical coordinates and you will obtain only r hat term--> central force)
Re: many particle system..
The notation d/dr is an alternative way of writing the del operator.
d/dr = id/dx +jd/dy + kd/dz
As for the first part of your question let us denote the displacement vector r_i - r_j (note that this is NOT the same as |r_i - r_j|!) by r for convenience. If your potential is a function of |r| and does not depend on theta and phi then to show the force is central simply take the negative gradient:
-grad V(r) = -rdV/dr since the potential depends on r only. From this you can see that the force acts in the direction of r which was the vector connecting the centers of the two interacting bodies. It follows that the force is central.
d/dr = id/dx +jd/dy + kd/dz
As for the first part of your question let us denote the displacement vector r_i - r_j (note that this is NOT the same as |r_i - r_j|!) by r for convenience. If your potential is a function of |r| and does not depend on theta and phi then to show the force is central simply take the negative gradient:
-grad V(r) = -rdV/dr since the potential depends on r only. From this you can see that the force acts in the direction of r which was the vector connecting the centers of the two interacting bodies. It follows that the force is central.