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MathGroup Archive 2006

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vector derivatives

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70289] vector derivatives
  • From: "rych" <rychphd at gmail.com>
  • Date: Thu, 12 Oct 2006 05:36:46 -0400 (EDT)

I have (n) particles {ri} (ri- radius-vectors) with pairwise
interaction potential (u) depending on the distance only
(|rij|=|ri-rj|) (in d=3 Euclidean space). I'd like Mathematica to find
the force, f_i and higher partial derivatives of the potential energy
U:
U = 0.5 \sum_i,j u(|rij|)
-fk= \nabla_k U= \sum_j u`(|rkj|) rkj/|rkj|
\nabla . \nablaU = \sum_i,j u``(|rij|)+(d-1)u`(|rij|)/|rij|
and so on.

And I'd like to have the results in that compact form. I start writing
an exersise like this in Mathematica, - not pretty at all and the
output is far from what I want.
r1 = {a1, b1, c1}; r2 = {a2, b2, c2};
l[r12_List] := Sqrt[r12 . r12]
u[r1_List, r2_List] := l[r1 - r2]^n
D[l[r1 - r2], r1]
D[u[r1, r2], r1]

What is the proper way of doing such a task in Mathematica (and in
mathematics)? If the Tensorial package is the way to go, I don't mind
learning it, metric tensors, differential forms and such.
Thanks
Igor


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