(2006-11-12) Clausius' Virial of Force (1870)
A classical quantity homogeneous with an energy.

The name virial was coined in 1870
by Rudolf Clausius (1822-1888) from the genitive viris
of the Latin word vis (force, power or energy).

In the classical study of a system of N point-masses,
Clausius' original virial of force,
is obtained by summing up, for each point-mass, the dot
product of its position by the force it's subjected to:

N

å

F_{i} . r_{i}

i = 1

In the absence of external forces, the sum
å_{i}F_{i} vanishes,
so the above doesn't depend on the arbitrary choice of an origin for spatial positions.

(2006-11-11) Classical Virial
Unless otherwise specified, "virial" means "virial of momentum"...

Consider the sum of the scalar products of the momenta and spatial positions:

G = ^{ }

N

å

p_{i} . r_{i}

i = 1

The total momentum p =
å_{i}p_{i} vanishes
if the observer is at rest with respect to the system's center of mass O.

For such an observer, the above expression does not depend
on the choice of the origin for positions in space. It's thus
called the proper virial G_{o}.

Otherwise, we may rewrite the above in terms of the total momentum
p = Mv,
the position r of the center of mass O,
and the position r_{oi} of i relative to O.

G =
å_{i}p_{i} . ( r + r_{oi })
^{ } = p . r +
å_{i}
m_{i} ( v + v_{oi }) . r_{oi}

G = p . r + G_{o}

The virial is
half the time-derivative of the moment of
inertia with respect to a point.
The moment of inertia about the center of mass O
is itself half of the trace of the
matrix of inertia.

(2006-11-12) Relativistic Virial
We must consider particles at constant time
in the observer's frame.

Relativistic definitions
for spatially extended bodies are delicate
because simultaneity depends on the motion of the observer.
Even the best authors have butchered the subject at times...

The notion of spacetime coordinates examined with extreme care by Einstein
has one overwhelming quality: It's consistent.
Maxwell's equations require such a notion.

Paradoxes would soon arise if we were to define some observed quantities
with one definition of simultaneity and other related quantities with another.

(2006-11-12) The quantum virial
The quantum counterpart of the classical virial.