Coleman Lecture: Varying E-M Lagrangian - Problem 3.1 Explained

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In summary, Coleman explains the use of integration by parts in deriving the second equation in problem 3.1. This can be simplified by writing the equations $\delta \mathcal{L}=-\frac{1}{4} \delta (F_{\mu \nu} F^{\mu \nu})$ and $\delta I = -\int \mathrm{d}^4 x \partial_{\mu} \delta A_{\nu} F^{\mu \nu}$, which leads to the free Maxwell equations for the potential, $\partial_{\mu} F^{\mu \nu}=0$ and $F_{\mu \nu}=\partial_{\mu} A_{\nu} -
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This is from Coleman Lectures on Relativity, p.63. I understand that he uses integration by parts, but just can't see how he gets to the second equation. (In problem 3.1 he suggest to take a particular entry in 3.1 to make that more obvious, but that does not help me.)
 
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You can simplify the task a bit by writing
$$\delta \mathcal{L}=-\frac{1}{4} \delta (F_{\mu \nu} F^{\mu \nu}) = -\frac{1}{2} \delta F_{\mu \nu} F^{\mu \nu}=-\delta (\partial_{\mu} A_{\nu}) F^{\mu \nu}.$$
Then you have, indeed via partial integration)
$$\delta I = -\int \mathrm{d}^4 x \partial_{\mu} \delta A_{\nu} F^{\mu \nu} = + \int \mathrm{d}^4 x \delta A_{\nu} \partial_{\mu} F^{\mu \nu} \stackrel{!}{=}0,$$
and from this you get the free Maxwell equations
$$\partial_{\mu} F^{\mu \nu}=0, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}$$
for the potential.
 
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FAQ: Coleman Lecture: Varying E-M Lagrangian - Problem 3.1 Explained

What is the Coleman Lecture about?

The Coleman Lecture is about the varying E-M Lagrangian and its application to Problem 3.1. This lecture explains the mathematical concepts and calculations involved in solving this problem.

What is the E-M Lagrangian?

The E-M Lagrangian is a mathematical function that describes the dynamics of electromagnetic fields. It is used in the study of electromagnetism and is an important tool in theoretical physics.

What is Problem 3.1?

Problem 3.1 is a specific mathematical problem that involves the varying E-M Lagrangian. It is often used as an example to demonstrate the application of this concept in solving real-world problems.

Why is understanding the varying E-M Lagrangian important?

Understanding the varying E-M Lagrangian is important because it allows scientists to accurately describe and predict the behavior of electromagnetic fields. This is crucial in many fields of physics, such as quantum mechanics and cosmology.

What can I expect to learn from the Coleman Lecture?

The Coleman Lecture will provide a detailed explanation of the varying E-M Lagrangian and its application to Problem 3.1. It will also cover the underlying mathematical concepts and equations involved, giving you a better understanding of this important topic in theoretical physics.

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