Minimizing an Action Integral: Solving for G(z)

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In summary, the problem involves finding the path of a light ray in a medium with varying dielectricity constant between two points A and B. The action integral is minimized and is equivalent to a differential equation, which can be solved to find the general form for the light ray's path. However, the notation and equations provided may not be correct.
  • #1
JohanL
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If you have an action integral

[tex]
\int_{A}^{B} \sqrt{F\mathbf{(r)}} dr
[/tex]

and

F=a-bz^2 , b>0, a-bd^2>0

the minimum of the action integral is equivalent to

[tex]
\frac{d}{dt}\frac{dG}{\dot{z}}-\frac{dG}{z}=0
[/tex]

where
[tex]
G=\sqrt{F}
[/tex]

or am i doing this in a completeley wrong way?
 
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  • #2
You jump around a lot with your notation. See if this is clear:

If you have an Integral of this form(where y' is the derivative of y with respect to x):

[tex] \Int F[y , y' , x ] dx [/tex]

Then the function y which minimizes the integral is the solution to this differential equation:

[tex] \frac {d}{dx} (\frac {\partial F}{\partial y'}) - \frac{\partial F}{\partial y} = 0 [/tex]
 
  • #3
Extremizes...

Daniel.
 
  • #4
It was those equations i was trying to use...but probably in a wrong way.
To explain my notation i give you the problem:

A light ray's path in a medium with variating dielectricity constant

[tex]
\epsilon(\vec{r})
[/tex]

between the two points A and B, minimizes, according to Fermat's principle, the action integral

[tex]\int_{A}^{B} \sqrt{\epsilon(\vec{r})} |d\vec{r}| [/tex]

A plane piece of glass with thickness 2d has a dielectricity function which, when the piece is in |z|<=d in a cartesian coordinatesystem, can be written

[tex]
\epsilon(\vec{r})=a-bz^2 ,b>0, a-bd^2>0

[/tex]

Calculate the general form for the light ray's path in the piece of glass.

Solution:

The minimum of the action integral is equivalent to
(But this is probably wrong)

[tex]\frac{d}{dt}\frac{d(\sqrt{\epsilon(\vec{r})} )}{\dot{z}}-\frac{d(\sqrt{\epsilon(\vec{r})} )}{z}=0[/tex]

and with this you get

-1/2(a-bz^2)^(-1/2) * 2bz = 0

and this must be wrong!
 

1. What is an action integral?

An action integral is a mathematical concept used in physics and engineering to describe the total amount of action, or energy, required for a system to transition from one state to another over a given time period.

2. What does it mean to minimize an action integral?

Minimizing an action integral means finding the path or trajectory that a system must follow in order to use the least amount of energy to transition from one state to another. This is often referred to as the "principle of least action."

3. How does minimizing an action integral relate to solving for G(z)?

Minimizing an action integral is a crucial step in solving for G(z), which is the Green's function in the calculus of variations. The Green's function is a mathematical tool used to solve for the path that minimizes the action integral for a given system.

4. What techniques are used to minimize an action integral?

The most common technique used to minimize an action integral is the Euler-Lagrange equation, which involves taking the derivative of the action integral with respect to the system's variables and setting it equal to zero. Other techniques, such as the Hamilton-Jacobi method, can also be used.

5. Why is minimizing an action integral important?

Minimizing an action integral is important because it allows us to understand the fundamental principles of a system and predict its behavior. It is also a useful tool in various fields of science and engineering, such as classical mechanics, electromagnetics, and quantum mechanics.

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