# Integral in a variational principle problem

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• MrMuscle
In summary, if you want to solve the integral by yourself, you need to look inside the back cover of the book.
MrMuscle
TL;DR Summary
Trying to solve the integral, for variational principle in griffith's book.
Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!

MrMuscle said:
Summary: Trying to solve the integral, for variational principle in griffith's book.

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!
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You could make ##e^{-2bx^2}## look like a Gaussian zero-mean probability density function and use known results for the area under the curve and the variance.

MrMuscle said:
Summary: Trying to solve the integral, for variational principle in griffith's book.

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't go further. A little bit help to finish the integration would do great! Thanks for your help in advance!
View attachment 253010
View attachment 253009
If you look inside the back cover of the Griffiths book you might find those standard integrals.

If you want to derive them yourself, the first can be done by a clever trick and transforming to polar coordinates; and the second can be reduced to the first using integration by parts.

PS I'm not sure how you got so far in the book without doing those integrals about 20 times already!

## 1. What is the significance of the integral in a variational principle problem?

The integral in a variational principle problem represents the total energy or action of a system. It is used to find the optimal solution that minimizes or maximizes this energy or action.

## 2. How is the integral calculated in a variational principle problem?

The integral is calculated by taking the integral of the Lagrangian function, which is a combination of the system's kinetic and potential energies. This integral is then minimized or maximized using the Euler-Lagrange equation to find the optimal solution.

## 3. What is the relationship between the integral and the Euler-Lagrange equation?

The Euler-Lagrange equation is used to find the stationary points of the integral. These stationary points correspond to the optimal solution that minimizes or maximizes the integral, thus solving the variational principle problem.

## 4. Can the integral be used to solve any type of variational principle problem?

Yes, the integral can be used to solve a wide range of variational principle problems in physics, mathematics, and engineering. It is a powerful tool for finding optimal solutions in various fields.

## 5. How does the integral relate to the principle of least action?

The integral in a variational principle problem is closely related to the principle of least action, which states that the actual path taken by a system is the one that minimizes the total action. The integral is used to calculate this total action and find the optimal path or solution.

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