Why is the first term zero in the Virial Theorem derivation?

In summary, the conversation discusses the derivation of the Virial Theorem and a step that raises questions. The question is about how Newton's Third Law affects the equation 2.41 and how the symmetry is involved. It is explained that the two terms with interchanging indices cancel each other out, leaving only the diagonals which are zero due to a particle not exerting a force on itself. However, it is clarified that the sum is not over the entire vector F_ij, but only the component along the position vector of the central point. It is then questioned why the second term is not also zero, to which it is explained that the vector changes sign when the indices are exchanged.
  • #1
rexregisanimi
43
6
Hello everyone! I am reviewing the derivation of the Virial Theorem from an introductory Astrophysics book (Carroll and Ostlie's) and found a step I couldn't follow. I've attached a photo of the step.

Can anyone explain how Newton's Third Law brings about eqn 2.41? I don't see how that first term in the right side previous to eqn 2.41 goes to zero. What symmetry is being referenced?

1482184447251.jpg
 
Last edited:
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  • #2
Consider the ##ij##-th term
$$
\mathbf F_{ij} \cdot (\mathbf r_i + \mathbf r_j)
$$
and the ##ji##-th term
$$
\mathbf F_{ji} \cdot (\mathbf r_j + \mathbf r_i) = -\mathbf F_{ij} \cdot (\mathbf r_i + \mathbf r_j)
$$
The two terms has equal magnitude but opposite sign, so they will cancel. The same is true for other pair of terms connected by interchanging the indices.
 
  • #3
So, in other words, the top triangular part of the F_ij matrix cancels with the bottom triangular part because of the Third Law; this leaves the diagonals only which are zero because a particle will not exert a force on itself. Is that correct?
 
  • #4
You actually don't sum the vector F_ij, instead you are summing over the component of F_ij along the position vector of the central point in the line connecting the two particles.
 
  • #5
Thank you for your replies.

Why then isn't the second term zero? If we're just concerned about those values, shouldn't both terms go to zero?
 
  • #6
rexregisanimi said:
Thank you for your replies.

Why then isn't the second term zero? If we're just concerned about those values, shouldn't both terms go to zero?
When you exchange the indices, the vector ##\mathbf r_i - \mathbf r_j## also changes sign.
 

1. What is the Virial Theorem?

The Virial Theorem is a mathematical relationship that describes the average kinetic and potential energy of a system in equilibrium. It states that the total kinetic energy of a system is equal to the negative of half the total potential energy.

2. How is the Virial Theorem derived?

The Virial Theorem is derived using the principle of virialization, which states that in a system of particles that interact through a central force, the average value of the radial component of the force is equal to the average value of the product of the radial position and the force. This principle is applied to the equations of motion of the particles, resulting in the Virial Theorem.

3. What is the significance of the Virial Theorem?

The Virial Theorem is significant because it allows us to understand the relationship between the kinetic and potential energy of a system in equilibrium. It also has applications in a wide range of fields, including astrophysics, molecular dynamics, and fluid mechanics.

4. Can the Virial Theorem be extended to non-equilibrium systems?

Yes, the Virial Theorem can be extended to non-equilibrium systems by taking into account the time derivative of the average value of the force. This results in a generalized version of the theorem that is applicable to a wider range of systems.

5. Are there any limitations to the Virial Theorem?

Yes, the Virial Theorem is limited in its applicability to systems that obey the conditions of virialization, such as having interactions that follow an inverse square law and being in a state of equilibrium. It also does not take into account the effects of external forces or non-conservative forces.

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