How does the Virial Theorem derivation for <K>=-<delta(U)> work?

In summary, the viral theorem is a mathematical principle that relates the total kinetic and potential energies of a system over time. It is based on the concept of scalar moment of inertia and the first derivative, known as the virial. The theorem states that in a stationary state, the time averages of kinetic and potential energies are equal to each other, with the potential energy being a power law. This theorem can be applied in astronomy to measure the total energies of a system at a given time, with the assumption that the system is already virialized. More information on the theorem can be found on Wikipedia.
  • #1
Himanshu
67
0
Can somebody explain me how does the derivation for <K>=-<delta(U)> goes under the viral theorem?
 
Astronomy news on Phys.org
  • #2
1.They define something called called 'scalar moment of inertia around the origin', I. It looks similar to moment of inertia around an axis in mechanics but not quite.

2. The first derivative G = (1/2) dI/dt is called 'virial'. They prove that the first derivative of the virial, dG/dt, i.e. the second time derivative of I , depends on the total kinetic energy T of the system and the potential energy U, when the potential energy is a power law ~ r^n:

dG/dt = 2T - nV (n=-1 for gravity)

3. They take the time average <...> of the above equation and claim that after long time the system 'virializes' so that the time average <dG/dt> = 0 which gives you an equation between the time averages of kinetic energy and potential energy:

<T> = (n/2) <V>

4. If we assume the system is 'virialized' i. e. in a stationary state, equilibrium, so that the total kinetic and potential energy do not change with time, the time averages will equal the energies at any time.

When that theorem is applied in astronomy, we measure/observe only the total kinetic and potential energies at a given time. We don't have access to time averaged values because the times involved are millions/billions of years but we assume the system is virialized already so the energy values equal the time averages.

All that (except point 4) can be found here http://en.wikipedia.org/wiki/Virial_theorem
 
Last edited:
  • #3
Thanks!
 

What is the Virial Theorem derivation?

The Virial Theorem derivation is a mathematical equation that relates the average kinetic energy of a system to its potential energy. It is used in physics and astronomy to understand the behavior of systems, such as galaxies or particles in a gas.

How is the Virial Theorem derived?

The Virial Theorem is derived using Lagrangian mechanics, which is a mathematical framework for analyzing the motion of systems. It involves applying the principle of virtual work to the system and solving for the equations of motion.

What are the assumptions made in the Virial Theorem derivation?

The Virial Theorem derivation assumes that the system is in a steady state, meaning that it is not changing over time. It also assumes that the system is isolated, meaning that there are no external forces acting on it.

What is the significance of the Virial Theorem derivation?

The Virial Theorem derivation is significant because it allows for the calculation of the average kinetic energy of a system, which is important in understanding the behavior of physical systems. It also has applications in astrophysics, such as in the study of star formation and galactic dynamics.

Are there any limitations to the Virial Theorem derivation?

Yes, there are limitations to the Virial Theorem derivation. It assumes that the system is in equilibrium, which is not always the case in real-world systems. It also does not take into account the effects of external forces, such as friction or drag, which can impact the behavior of a system.

Similar threads

Replies
2
Views
2K
  • Astronomy and Astrophysics
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
731
  • Astronomy and Astrophysics
Replies
2
Views
2K
  • Astronomy and Astrophysics
Replies
3
Views
2K
  • Astronomy and Astrophysics
Replies
1
Views
3K
  • Introductory Physics Homework Help
Replies
11
Views
903
  • Classical Physics
Replies
2
Views
806
  • Astronomy and Astrophysics
Replies
1
Views
4K
Replies
5
Views
2K
Back
Top