# Variation principle -- looking for resources to read and understand

• Pouramat
In summary, there are multiple resources available for learning the variation principle, including textbooks such as "Spacetime Geometry" by Sean Carroll, "Emily Noether's Wonderful Theorem" by Neuenschwander, "SR and Classical Field Theory" by Susskind, "Elementary Particles" by Griffiths, "QFT for the Gifted Amateur" by Lancaster & Blundell, and "QFT in a Nutshell" by Zee. It is recommended to first study the action principle in classical mechanics and then classical field theory before delving into quantum field theory. One suggested starting point is Landau and Lifshitz volumes 1 and 2. However, it is advised to avoid Zee's book as
Pouramat
Summary:: Can anyone introduce an informative resource with solved examples for learning variation principle?

For example I cannot do the variation for the electromagnetic lagrangian when ##A_\mu J^\mu## added to the free lagrangian and also some other terms which are possible:
$$L = \sqrt{-g}(\frac{-1}{4}F^{\mu \nu}F_{\mu \nu}+A_\mu J^\mu)$$
also I want to add ##L' = \beta R^{\mu \nu} g^{\rho \sigma}F_{\mu \rho}F_{\nu \sigma}## to lagrangian and then do the variation to see what is the difference, but I cannnot.
Do you know any good resources to understand with solved examples?

How have you got this far in your physics studies without learning the Lagrangian formulation and variational principle?

I must have at least four text-books that cover this in one form or other. E.g. Spacetime Geometry, by Sean Carroll.

PeroK said:
How have you got this far in your physics studies without learning the Lagrangian formulation and variational principle?

I must have at least four text-books that cover this in one form or other. E.g. Spacetime Geometry, by Sean Carroll.
Can you give me another reference except Carroll?

Pouramat said:
Can you give me another reference except Carroll?
Emily Noether's Wonderful Theorem - Neuenschwander

SR and Classical Field Theory (The Theoretical Minimum) - Susskind

Elementary Particles - Griffiths

QFT for the Gifted Amateur - Lancaster & Blundell

QFT in a Nutshell - Zee

Pouramat
I would first study the action principle within classical mechanics and then classical field theory before I'd go into quantum field theory. For this I'd recommend Landau and Lifshitz vols. 1 and 2.

If you start with the latter, don't take Zee's book. It's fun to read if you know the subject from somewhere else. It's not a good source to start learning it, because it wants to put too much in the nutshell and is thus too superficial to explain QFT for a beginner. Lancaster and Blundell is way better.

Pouramat

## What is the variation principle?

The variation principle is a fundamental concept in science that states that all natural phenomena exhibit variation, or differences, within a population. This variation is the result of genetic and environmental factors and is essential for evolution and adaptation.

## How is the variation principle important in scientific research?

The variation principle is crucial in scientific research as it allows scientists to understand and study the diversity of life on Earth. By recognizing and studying variations within a population, scientists can better understand the underlying mechanisms that drive biological processes and make predictions about future changes.

## What are some examples of variation in nature?

Examples of variation in nature include differences in physical traits among individuals, such as height or eye color, as well as variations in behavior, such as mating rituals or foraging strategies. Additionally, genetic variations within a population can lead to differences in susceptibility to diseases or environmental stressors.

## How does the variation principle relate to evolution?

The variation principle is closely tied to the concept of evolution. Natural selection, a key mechanism of evolution, acts on the variations within a population, favoring traits that provide a survival or reproductive advantage. Over time, these favorable traits become more common in the population, leading to evolutionary change.

## Where can I find more resources to learn about the variation principle?

There are many resources available to learn about the variation principle, including textbooks, scientific articles, and online resources from reputable organizations and institutions. Some recommended sources include the National Center for Biotechnology Information, the National Science Foundation, and the National Academy of Sciences.

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