The Virial Theorem is an important theorem in mechanics. It is marvelously simple to prove. In the material below the variables which are vectors will be displayed in red.

Consider a system of n point particles indexed by i. Let r_i, v_i and p_i be the position, velocity and momentum vectors, respectively, for the i-th particle. Its mass is denoted as m_i. Let F_i be the net force, internal and external, impinging upon the i-th particle.

• Statement of the Virial Theorem: For the n point particles bound together into a system the time average of the kinetic energy of the particles, Σ½m_iv_i², plus one half of the time average of ΣF_i·r_i, the virial of the particles, is equal to zero.

Proof

For each particle

v_i = dr_i/dt
p_i = m_iv_i
dp_i/dt = F_i

Define H as Σp_i·r_i. (Note that the dot product of two vectors is a scalar.) The quantity H does not have a generally accepted name, although it seems it should have. It is in the nature of a moment.

Then

dH/dt = Σ(dp_i/dt)·r_i + Σ p_i·(dr_i/dt)

Because (dp_i/dt) = F_i the first term on the right in the above equation reduces to ΣF_i·r_i. This quantity was called the virial of the system by Clausius in 1870, a term he coined based upon the Latin term vis, meaning force or energy.

Because p_i=m_iv_i =m_i(dr_i/dt) the second term on the right in the previous equation reduces to

Σ p_i·(dr_i/dt) = Σ m_i(dr_i/dt)² =Σm_iv_i²

This last expression is just twice the kinetic energy K of the system; i.e., 2Σ½m_iv_i². Thus

dH/dt = ΣF_i·r_i + 2K

The Time Average of the Variables

The time average of a variable y(t) over the interval 0 to τ is defined as

y = (1/τ)∫₀^τ y(t)dt

Time averaging the equation of dH/dt gives

dH/dt = ΣF_i·r_i + 2K.

The time average of (dH/dt) is just

dH/dt = (1/τ)[H(τ)−H(0)]

If the system is cyclical such that it returns to its initial state after an interval then τ can be chosen equal to the cycle period and dH/dt reduces to 0. If the system is not cyclical then for a system which is bounded the limit of dH/dt as τ increases without bound is zero.

Thus

ΣF_i·r_i + 2K = 0
or, equivalently
 
K + ½ΣF_i·r_i = 0

The Nature of Vectors

A vector is defined by three elements; its location, its direction and its magnitude. Vectors are often represented as arrows. The origin of the arrow specifies its location, the direction of the arrow is the direction of the vector and the length of the arrow is the magnitude of the vector. The subtle problem in the virial theorem is the specification of the origins for the vectors. In case of the forces, location of a force on a particle is simply the location of the particle. Forces cannot impinge upon the system of particles except through the particles.

The location of the origin for the position vectors, r_i, is a different matter. The origin for the position vectors would seem to be entirely arbitrary. However a change in the origin will affect the virial by changing the magnitudes of the position vectors and also through changing the angles between the position vectors and the force vectors.

Suppose that for one specification of the origin of the position vectors the value of the H is determined

H = Σ p_i·r_i

Then suppose the origin of the position vectors is moved and let r_oo represent the movement of the origin. The new position vectors are then given by

r'_i = r_i + r_oo

The forces within a system typically depend only upon the distances between the particles and these distances are not affected by a change in the origin of the position vectors. Likewise the momenta and kinetic energies would seem to be not affected by a change in the origin of the position vectors. But the values of H and ΣF_i r_i would seem to be affected.

The new value of H' is then

H' = Σ p_i·r'_i
H' = Σ p_i·(r_i+r_oo)
H' = H + Σ p_i·r_oo

Likewise

ΣF_i·r'_i = ΣF_i·r_i + ΣF_i·r_oo

The origin of the position vectors could be located at the center of the Andromeda Galaxy, which is 2.5 million light years away, thus seemingly affecting the value of H and ΣF_i·r_i enormously .

If the change of origin were to change the H only by a constant amount then the dynamics of the system would be unaffected. However by the virial theorem the average kinetic energy of the system would change by the negative of half the amount of the change in the virial. This would seem to indicate that the origin of the position vectors cannot be chosen arbitrarily, even though the proof of the virial theorem does not seem to impose any restriction on that origin.

It is known that the virial theorem holds for the choice of the origin of the position vectors being at the center of mass of the system. To see that the virial theorem holds for other choices for the origin of the position vectors consider the case for a two particle system.

Two Particle Systems

Consider a two-particle system held together by gravitation and that the orbits of the particles are circles with their centers at the center of mass of the particles. Furthermore assume the masses of the particles are both m. If s is the separation distance then the orbits have a radius of s/2. Then the gravitational attraction must be sufficient to generate the centripetal acceleration necessary to keep the particles in circular orbits. This requires that

mv²/(s/2) = Gm²/s²

where v is the tangential velocity and G is the graviational constant.

The kinetic energy of one particle is ½mv² so the total kinetic energy of the system, K, is mv². From the above relation

K = mv² = ½Gm²/s

The RHS of this equation happens to be one half of negative of the potential energy V of the system, so

K = −½V
or, equivalently
K + ½V = 0

The position vectors with respect to the center of mass have magnitudes equal to (s/2). They are parallel with the forces and point in the opposite direction. Thus H of the system is given by

ΣF_ir_i = −2(Gm²/s²)(s/2) = −Gm²/s

Thus the virial theorem is exactly satisfied.

Now assume the origin of the coordinate system is at the center of one of the particles, say Particle 1. Then r₁ is equal to zero and r₂ is equal in magnitude to the separation distance for Particle 2.

The force between the particles is given by

F_i = −Gm₁m₂/|r₂-r₁|²
which reduces to
F_i = −Gm₁₂/|r₂|² = −Gm²/s²

The value of H is then given by

H = −(Gm²/s²)s = −Gm²/s

Again the virial theorem is exactly satisfied.

The Effect of a Change in the
Origin of the Coordinate System

Consider the position vectors for one origin r and another origin r'. (Here the prime (') denotes a different case rather than differentiation.) Then

r'(t) = r(t) + r_oo(t)

where r_oo(t) is the vector between the origin of the first coordinate system and the origin of the second coordinate system. The velocities are then related by

dr'/dt = dr/dt + dr_oo/dt
or, equivalently
v'(t) = v(t) + v_oo(t)

If the origins of the position vectors are motionless then v'(t)=v(t), and likewise the momenta vectors, p'(t)=p(t). However this is not the case with the value of p(t)·r(t).

p'(t)r'(t) = p(t)·r(t) + p(t)·r_oo(t)

For forces depending only upon the separation of the particles F'(t)=F(t). But

F'(t)·r'(t) = F(t)·r(t) + F(t)·r_oo(t)

For summation over all the particles of the system then

ΣF'_i(t)·r'_i(t) = ΣF(t)·r(t) + (ΣF_i(t))·r_oo(t)

For systems subject only to internal forces the net force is zero because the forces between any two particles cancel out. (Newton's third law of mechanics.) Thus the virials for the systems under the two coordinate systems are the same. Thus if (but not only if) the origins of the coordinate systems are motionless and the particles are subject only to internal forces dependent only upon the separation of the particles the virial theorem holds for arbitrary locations of the origin of the coordinate system.

Conclusion

For systems of particles subject only to internal forces the location of the origin of the coordinate system does not affect the value of ΣF_i·r_i and the virial theorem holds regardless of the location of the origin of the coordinate system.

(To be continued.)