# Obtaining field equations from an action

• hepnoob92
In summary, the conversation discusses a provided action and how to find the field equations for it using the Euler-Lagrange method. The question also asks for resources that can help with understanding the notation and extracting the physics from the action. The response suggests starting with the EL equation for a single component of the field.
hepnoob92

## Homework Statement

Provided an action:
$$S[A_\nu] = \int\left(\frac{1}{4}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta A_\mu J^\mu\right)\sqrt{-\eta}~d^4x$$

How would one go about finding the field equations for the same? I do understand that using the Euler-Lagrange method is how one should start out.

Does it tell us what kind of field we're looking at, at a glance?

## The Attempt at a Solution

I know that the terms in the parentheses are just ##F_{\mu\nu}F^{\mu\nu}##, but am unsure of how to proceed.

Does anyone know any good resources online or books that derive field equations and extract physics from a given action in steps, giving a good detailed process from which one can observe and learn how the Euler Lagrange method is being applied, in terms of the 4-vector notation. My struggle is essentially in decoding the physics, largely due to problems with understanding the notation.

Last edited:
hepnoob92 said:
I do understand that using the Euler-Lagrange method is how one should start out.
Right, so what do you get when you do the EL equation for a single component ##A_\lambda## of the field?

## 1. What is the action principle?

The action principle is a fundamental concept in physics that states that the behavior of a physical system can be described by minimizing an action, which is a mathematical quantity that represents the total energy of the system over a period of time. This principle is used to derive field equations, which are equations that describe the behavior of a field (such as electromagnetic or gravitational fields) in terms of its sources.

## 2. How are field equations obtained from an action?

To obtain field equations from an action, one must first choose a suitable action that describes the system in question. This is often done by considering the symmetries and conservation laws of the system. Then, variations of the action with respect to the fields in the system are taken, resulting in a set of Euler-Lagrange equations. These equations are then solved to obtain the desired field equations.

## 3. What are the advantages of using the action principle to obtain field equations?

The action principle provides a unified and elegant approach to describing physical systems, as it allows for the derivation of field equations from a single action. It also takes into account the symmetries and conservation laws of the system, making the resulting equations more physically meaningful. Additionally, the action principle is often more mathematically tractable and can lead to simpler and more intuitive equations.

## 4. What are some examples of field equations obtained from an action?

Some well-known examples of field equations obtained from an action include Maxwell's equations, which describe the behavior of electromagnetic fields, and Einstein's field equations, which describe the behavior of gravitational fields in general relativity. Other examples include the Klein-Gordon equation, which describes the behavior of a scalar field, and the Dirac equation, which describes the behavior of a spinor field.

## 5. Are there any limitations to using the action principle to obtain field equations?

While the action principle is a powerful tool for obtaining field equations, it is not always applicable to all physical systems. In some cases, the action may be unknown or difficult to construct, making it impossible to use this method. Additionally, the resulting field equations may not always accurately describe the behavior of a system, as they are based on certain assumptions and approximations. Therefore, it is important to carefully consider the validity of using the action principle in each specific case.

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