Principle of Least Action in Electrostatics

[nassboy]

In Feynman's lecture on the principle of least action, he show that you can describe electrostatics by saying that a certain integral is a minimum or maximum. He states the integral, and goes on to show how it can be used to approximate the capacitance of a coaxial line. I've been able to follow his argument and use a similar technique to solve the parallel plate capacitor.

Later Feynman states that, "For any other shape, you can guess an approximate field with some unknown parameters like alpha and adjust them to get a minimum. You will get excellent numerical results for otherwise intractable problems."

I can't seem to apply his method to anything but the simplest of cases. For example, coming up with an approximate field that meets complicated boundary conditions in 2D seems impossible...leaving conformal mapping as the only usable method of solving such problems. I also tried solving another problem that Feynman solves in a more traditional manner, "Colloidal particles in an electrolyte." using the least action principle...but what approximate field should I choose besides a polynomial?

 

[gtoguy97]

I'm having trouble understanding what kind of field is best used in these cases. Can someone explain how the principle of least action can be used for more complex problems? Is there a general method of constructing an approximate field for a given problem? How do I choose the parameters such as alpha? Any help would be greatly appreciated!

 

[astrokreng]

The principle of least action in electrostatics is a powerful tool in solving complex problems and obtaining numerical results. However, it is important to keep in mind that it is not a one-size-fits-all approach and may not be applicable to all cases.

In the case of complex boundary conditions in 2D, it may be challenging to come up with an approximate field that meets all the conditions and still yields a minimum or maximum value for the integral. In such cases, other methods like conformal mapping may be more suitable.

When it comes to choosing an approximate field for a problem, it is important to consider the physical properties and boundary conditions of the system. A polynomial may work for some cases, but for others, a different functional form may be more appropriate. It may also be helpful to use a combination of different functions to approximate the field.

Overall, the principle of least action in electrostatics is a valuable tool, but it is important to consider its limitations and use it in conjunction with other methods for more complex problems. it is always important to explore and experiment with different techniques to find the best approach for a particular problem.