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[Phys-L] Re: PHYS-L Digest - 15 Mar 2005 to 16 Mar 2005 (#2005-93)



On 16-Mar-05, at 1:00 AM, Bob Sciamanda wrote (as corrected):

I would point out that the (mechanical) angular momentum of a system
can always be written as an "orbital " plus a "spin" term. The spin
term is truly a property of the system, since it is defined relative
to the center of mass as "origin". Further, one need not use the CM
for the origin of both positions (Ri) and velocities (Vi) in
calculating
L(spin) = L(cm) = SUM {Mi(Ri x Vi)}.

I am confused by this. The general definition of the angular momentum
of a system of particles with respect to *any* point is given by your
expression. It is only necessary that all the Ri be relative to that
same point. The Vi are merely time derivatives of the Ri; they are not
usually thought of as being related to an origin.

One may, of course, *define* the spin angular momentum as being the
angular momentum of a system of particles with respect to its center of
mass, but why should one do so?

It would seem that the spin angular momentum of the solar system by
your definition is what is usually referred to as the angular momentum
of the solar system with respect to its center of mass. In this case I
would find it confusing to refer to it as spin angular momentum. In
fact it is often imprecisely called, simply, the angular momentum of
the solar system.

You are correct, of course, in stating that the angular momentum of the
system with respect to any point P can be written as the sum of its
angular momentum with respect to its center of mass plus a term equal
to the angular momentum with respect to P of a particle having mass
equal to the system mass and velocity equal to the velocity of the
center of mass relative to P. It is this second term that you called
"orbital", I think. It could pertain to, for example, the motion of the
solar system with respect to the center of the Galaxy, in which case
the total angular momentum of the solar system with respect to the
center of the Galaxy would be a sum of such terms.

I dislike this use of "spin" and "orbital" nomenclature. Consider the
example of two solar systems bound in orbit with one another (a binary
stellar system) and you will see that the distinction is next to
useless.

In fact if one chooses one space point A for the origin of position
vectors and a second space point B for the origin of velocity vectors
(ie the origin for position vectors whose time derivatives are the
velocity vectors), one can show that the spin angular momentum is
independent of the choices for A and B, so long as one (or both) of
them is chosen to be the system center of mass. (IE, either the CM is
the origin of the Ri and/or the Vi are calculated relative to the CM
frame.)

I'm also confused by this.

The calculation of the angular momentum of a system of particles with
respect to a point is straightforward. It can be broken into two terms
(a theorem usually proved early on in introductory mechanics courses)
for ease of calculation, but no special status is accorded either term.
The principle of conservation of angular momentum must be applied to
the isolated system as a whole.

In systems of interacting particles of increasing complexity more
dynamical invariants arise. For such systems total energy and angular
momentum are not the only "integrals of the motion", as they are
referred to in the trade. For example, in an isolated three body
interacting particle system a quantity called "the Jacobi Integral" is
an invariant. See, e.g., C D Murray and S F Dermott, "Solar System
Dynamics".

Leigh
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