Directional Derivative of Potential energy

In summary, the conversation is about someone struggling to understand basic concepts of classical physics, specifically the meaning of "ij" in "Vij" and the equation F=-∇iVij. They are also confused about the use of ∇i instead of just ∇. The expert summarizer explains that "ij" represents particle numbers in the potential equation and ∇i refers to the change in position for particle i. They also clarify that the derivatives in the equation are with respect to the components of ri, not rj.
  • #1
vgptgupta
3
0
I'm facing some problem in understanding few basic concepts of classical physics.

http://www.fotoshack.us/fotos/67357p0020-sel.jpg I cannot understand what does "ij" indicate in "Vij" and how does F=-∇iVij. Why ∇i, why not only ∇.

Please help anybody. I'm practically getting frustrated googling for answers.

N.B. Equation 1.29
http://www.fotoshack.us/fotos/74847p0019-sel.jpg
 
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  • #2
i and j in Vij are indices for particle numbers. As an example, V23 is the potential between particle 2 and particle 3.
##\nabla_i## refers to the change of the position of particle i.
 
  • #3
Thanks for the "ij" explanation but what about my other question i.e., why F=-∇iVij and why ∇i, why not only ∇.
 
  • #4
There is no obvious meaning of ∇. Derivative for what?
What is meant here is the change in potential for particle i.
 
  • #5
So what does the line "the subscript i on the del operator indicates that the derivatives are with respect to the components of ri" mean.
 
  • #6
It means that the derivatives are with respect to xi, yi and zi, not xj, yj and zj.
 

FAQ: Directional Derivative of Potential energy

What is the directional derivative of potential energy?

The directional derivative of potential energy is a measure of the rate of change of potential energy in a specific direction. It is a vector quantity that represents the change in potential energy per unit distance along a specified direction.

How is the directional derivative of potential energy calculated?

The directional derivative of potential energy is calculated using the partial derivative of potential energy with respect to each coordinate direction multiplied by the corresponding direction vector.

What is the physical significance of the directional derivative of potential energy?

The directional derivative of potential energy is an important concept in physics, particularly in the study of conservative force fields. It helps us understand the change in potential energy as an object moves in a particular direction, and can be used to determine the direction of maximum increase or decrease in potential energy.

How is the directional derivative of potential energy used in real-world applications?

The directional derivative of potential energy is used in various fields such as mechanics, electromagnetism, and thermodynamics. It is used to calculate the work done by conservative forces, determine the stability of equilibrium points, and analyze the behavior of systems in different directions.

What are some limitations of the directional derivative of potential energy?

The directional derivative of potential energy assumes that the force field is conservative, meaning that the work done by the field is independent of the path taken. It also assumes that the potential energy is a continuous function, which may not always be the case in real-world systems.

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