Israel Khan Derivation Step: From Eq. 2.9 to 2.10

In summary, the conversation discusses the process of going from equation 2.9 to 2.10 in the classic paper "Collinear Particles and Bondi Dipoles in General Relativity" by Israel and Khan. The Bondi dipole moment is defined as the integral over the entire rod system of the product of the center of the rod and the mass of the rod. In equation 2.10, this integral is simplified using the definition of the center of mass of a system. This is possible because the Bondi dipole moment is a property of the entire rod system, not just individual rods. Further details can be found in the attached file.
  • #1
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I am looking at the classic paper Israel and Khan "Collinear Particles and Bondi Dipoles in General Relativity"

Nuovo Cimento 33 (1964) 331.

The basic question is how to go from equation 2.9 to 2.10?

a_i is the center of rod i and b_i is the mass of rod i.

The attached file has the details.

Thanks!
 

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  • #2


Hello,

Thank you for bringing up this interesting paper. I am familiar with it and I can help explain the process of going from equation 2.9 to 2.10.

In equation 2.9, we have the expression for the Bondi dipole moment, which is defined as the integral over the entire rod system of the product of the center of the rod (a_i) and the mass of the rod (b_i). This integral is taken over all the rods in the system, hence the summation over i.

In equation 2.10, we are essentially simplifying this integral by using the definition of the center of mass of a system. The center of mass is defined as the weighted average of the positions of all the particles in the system, where the weights are given by the masses of the particles. Mathematically, it can be expressed as the sum of the product of the positions and the masses, divided by the total mass of the system.

So, in equation 2.10, we are essentially replacing the integral over all the rods with the center of mass of the system, which is given by the term in parentheses. This simplification is possible because the Bondi dipole moment is a property of the entire rod system, not just individual rods.

I hope this helps to clarify the process of going from equation 2.9 to 2.10. Let me know if you have any further questions. Thank you.
 

1. How does the Israel Khan Derivation Step work?

The Israel Khan Derivation Step is a mathematical process used in control systems engineering to determine the stability of a closed-loop system. It involves taking the open-loop transfer function and adding a feedback term, which is multiplied by the original transfer function. This feedback term is then multiplied by a factor called the Israel Khan factor, which is determined by the system's poles and zeros. The result is a new transfer function, which can be used to analyze the stability of the closed-loop system.

2. What is the significance of Eq. 2.9 and 2.10 in the Israel Khan Derivation Step?

Eq. 2.9 and 2.10 refer to equations in the Israel Khan Derivation Step that represent the new transfer function after the feedback term and Israel Khan factor have been applied. These equations are important because they allow us to analyze the stability of the closed-loop system and make any necessary adjustments to ensure stability.

3. How do I know if my system is stable after using the Israel Khan Derivation Step?

The stability of a system can be determined by analyzing the poles of the new transfer function after the Israel Khan Derivation Step has been applied. If all of the poles are located in the left half of the complex plane, then the system is considered stable. If any poles are in the right half of the plane, the system is considered unstable and adjustments must be made.

4. Can the Israel Khan Derivation Step be used for any type of control system?

Yes, the Israel Khan Derivation Step can be applied to any type of control system that has a transfer function. This includes both continuous and discrete systems, as well as systems with multiple inputs and outputs.

5. How does the Israel Khan factor affect the stability of a closed-loop system?

The Israel Khan factor acts as a multiplier in the feedback term of the new transfer function. It is determined by the system's poles and zeros and can have a significant impact on the stability of the closed-loop system. If the Israel Khan factor is too large, it can cause the system to become unstable. Therefore, it is important to carefully consider the Israel Khan factor when using the Israel Khan Derivation Step.

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