Least Action Principle Applied to Vector Field ##A_{\mu}##

In summary, the conversation discusses a vector field with a Lagrangian of the form ##L=Const.{\times}F^{\mu\nu}F_{\mu\nu}##, where ##F_{\mu\nu}=({\partial}_{\mu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\mu})A_{\nu}-({\partial}_{\nu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\nu})A_{\mu}##. The least action principle is then applied to the Lagrangian, leading to a discussion on which of two choices for ##{\delta}S##
  • #1
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hello, everyone. When a vector field ##A_{\mu}## has the Lagrangian of the form as
##L=Const.{\times}F^{\mu\nu}F_{\mu\nu}##, where
[itex]F_{\mu\nu}=({\partial}_{\mu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\mu})A_{\nu}-({\partial}_{\nu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\nu})A_{\mu}[/itex]. Now I will apply the least action principle to it. Which one of the following two choices is the right one?
[itex]{\delta}S=\frac{{\partial}S}{{\partial}A_{\mu}}{\delta}A_{\mu}+\frac{{\partial}S}{{\partial}({\partial}_{\mu}A_{\nu})}{\delta}({\partial}_{\mu}A_{\nu})=0,[/itex]
or
[itex]{\delta}S=\frac{{\partial}S}{{\partial}A_{\mu}}{\delta}A_{\mu}+\frac{{\partial}S}{{\partial}({\partial}_{\mu}A_{\nu})}{\delta}({\partial}_{\mu}A_{\nu})+\frac{{\partial}S}{{\partial}({\partial}_{\mu}{\partial}_{\nu}{\partial}_{\rho}A_{\lambda})}{\delta}({\partial}_{\mu}{\partial}_{\nu}{\partial}_{\rho}A_{\lambda})=0.[/itex]
 
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  • #2
Which do you think and why?
 
  • #3
What is that alpha in your field strength? I'd say F=dA.

You can include higher derivatives in your functional derivative; if you use partial integrations, you'll see that every higher order derivative swaps a sign.
 
  • #4
haushofer said:
What is that alpha in your field strength? I'd say F=dA.

You can include higher derivatives in your functional derivative; if you use partial integrations, you'll see that every higher order derivative swaps a sign.
Thanks for your attention. I was reading a paper on a model of generalized uncertainty principle which originated from quantum gravity effect. ##\alpha## represents the parameter of quantum gravity effect.
 

What is the Least Action Principle Applied to Vector Field ##A_{\mu}##?

The Least Action Principle Applied to Vector Field ##A_{\mu}## is a fundamental principle in physics that states that a physical system will always follow the path of least action, or the path that minimizes the total energy expended.

How is the Least Action Principle Applied to Vector Field ##A_{\mu}## used in physics?

The Least Action Principle Applied to Vector Field ##A_{\mu}## is used to derive the equations of motion for a physical system. It can be applied to a wide range of systems, from classical mechanics to electromagnetism, and is a key tool in understanding the behavior of these systems.

What is the significance of the Least Action Principle Applied to Vector Field ##A_{\mu}##?

The Least Action Principle Applied to Vector Field ##A_{\mu}## is significant because it allows us to mathematically describe the behavior of physical systems. It is a fundamental principle in physics and is used to derive many important equations, such as Newton's laws of motion and Maxwell's equations.

How does the Least Action Principle Applied to Vector Field ##A_{\mu}## relate to other principles in physics?

The Least Action Principle Applied to Vector Field ##A_{\mu}## is closely related to other fundamental principles in physics, such as conservation of energy and momentum. It is also related to the principle of least time in optics, where light follows the path that minimizes the time taken to travel between two points.

What are some practical applications of the Least Action Principle Applied to Vector Field ##A_{\mu}##?

The Least Action Principle Applied to Vector Field ##A_{\mu}## has many practical applications in physics, such as predicting the motion of objects in space and understanding the behavior of electromagnetic fields. It is also used in various engineering fields, such as designing efficient pathways for fluid flow and optimizing energy usage in mechanical systems.

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