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(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	Fi . ri
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.

Wikipedia :   Virial theorem

(2023-07-03)     The equation of state of a gas can be derived with the virial theorem
Consider a mole of a homogenous gas in a container of volume V at pressure p and temperature T.

The only external forces applied to the gas are the pressure forces applied by the walls of the container.  Their virial is simply:

p V

The energy is the sum of the kinetic energy and the internal potential energy.

(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	pi . ri
	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  +   Go

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.