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Finalize Potential and Force

21/08/2011

Let x_i,x_i be two points. The potential between these two points is

V(x_i,x_j)

More prescisely, the potential actually is a function of x_i-x_j:

V(x_i-x_j)=V(x_j-x_i)

(Only depend on the distance but not direction, even function).

The force applied to x_i is F_{x_i}=-\nabla_{x_i}V(x_i,x_j).

The force applied to x_j is F_{x_j}=-\nabla_{x_j}V(x_i,x_j).

Highlight:

Physical interpretion: easy!  The direction of the force points to the negtive gradient!

The gradient of V(x_i-x_j) with respect to x_i means: if x_i can move (x_j can’t move), moving along the force direction, the potatial decrease the fastest.

Mathematically, it is easy to see \nabla_{x_i}V(x_i-x_j)=-\nabla_{x_j}V(x_i-x_j). Hence F_{x_i}=-F_{x_j}.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

EDIT:

The force and potential are only functions of x_i-x_j. More specifically, the potential is a function of \|x_i-x_j\|. So it can be usually seen that the force and potential are written as

F(x_{ij}) and V(x_{ij})

where x_{ij}=x_i-x_j.

And because \nabla_{x_{ij}}V(x_i-x_j)=\nabla_{x_{i}}V(x_i-x_j) and \nabla_{x_{ij}}V(x_i-x_j)=-\nabla_{x_{ji}}V(x_i-x_j)

The force applied to x_i is F_{x_{ij}}=-\nabla_{x_{ij}}V(x_i,x_j).

The force applied to x_j is $latex F_{x_{ji}}=-F_{x_{ij}}$.

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