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Let's now consider the
generalization of the result

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that the torque on a single
particle about a point

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causes the angular momentum of
that particle about that point

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to change to a
collection of particles.

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So let's begin by indicating
some ith particle with momentum

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Pi and some jth particle
over here with momentum Pj.

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And this is a big
system of n particles.

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And we'd like to
calculate the angular

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momentum about some point, s.

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So that angular momentum will
consist of the direct product

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of the vector from s
to the ith particle

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and the direct product of the
vector r s to the jth particle.

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So the angular
momentum, total, will

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be the sum over all
the particles, 1 to n,

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of this sum, rsj cross Pj.

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Now, what we'd like to
do is, again, as before,

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take the time derivative
of this angular momentum.

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We have the sum--

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j goes from 1 to n.

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We have two derivatives
here, by the product rule,

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cross Pj, plus the sum, j
goes from 1 to n of rsj,

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cross Fj, where we use the
fact, as we did before,

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that the force on
the jth particle

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is equal to the change in
momentum of the jth particle.

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Now, this first piece,
again, the derivative

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of the vector from s to the
jth part is the velocity.

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So we have j equals
1 to n of v--

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the velocity of
the jth particle--

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cross Mj, Pj.

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And that's 0, as we know
that a vector cross product

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with itself is 0.

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And this piece in here--

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now, we have to be a
little bit careful--

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but rsj cross Fj is the
torque on the jth particle.

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So that is the torque
on the jth particle.

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Now, remember that we showed
that internal-- the forces

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on the jth particle
could be both due

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to internal or external forces.

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And as long as the
internal forces

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pointed between two particles--

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pointed along the line
connecting those particles,

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the internal torques
cancel in pairs.

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And this will only be
the external torque.

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So we've assumed that
the internal torques

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cancel in pair.

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And that has to do with an
assumption about the direction

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of the internal forces.

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And so we can conclude
that this is just

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the total external torque
about the point s, s total.

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And we'll drop that
total, and we'll

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conclude that the torque
for our system of particles

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is just-- causes the angular
momentum of the system

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of particles to change.

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The calculation is exactly
like the single particle,

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with a few subtleties that have
to do with internal torques

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canceling in pairs.