# Variation of S with fixed end points

• Leb
In summary, the given notes explain the definition of a functional and its derivative. When the end points are fixed, the variation simplifies to the partial derivative of L with respect to y minus the derivative of L with respect to y'. It is not clear if this is the functional derivative or just delta S divided by delta gamma.
Leb
Not really a homework question, just the notes are confusing me.

## Homework Statement

Let S be a functional.

(Given without proof)
If S is differentiable its derivative $\delta S$ is uniquely defined as
$\delta S = \int_{x_{0}}^{x_{1}}\frac{\delta S}{\delta \gamma} \delta \gamma dx$ where $\frac{\delta S}{\delta \gamma}$

For functional S
$S[\gamma]=\int_{x_{0}}^{x_{1}}L(x,y(x),y'(x))dx$
the variation is defined by
$\delta S=\int_{x_{0}}^{x_{1}}\left( \frac{\partial{L}}{\partial{y}}-\frac{d}{dx}\frac{\partial{L}}{\partial{y'}}\right)\delta \gamma dx + \left.\frac{\partial{L}}{\partial{y'}}\delta \gamma \right|_{x_{0}}^{x_{1}}$

Define $\delta \gamma = \epsilon h(x)$ where $\epsilon =const.<< 1$ and h(x) is an arbitrary (perturbation) function.

Now, the notes given by the lecturer say, that if end points are fixed i.e. $h(x_{0})=h(x_{1})=0$ the variation simplifies to

$\frac{\delta S}{\delta \gamma}= \frac{\partial{L}}{\partial{y}}-\frac{d}{dx}\frac{\partial{L}}{\partial{y'}}$

I do not understand, how do we get this (I get how the last term in the variation vanishes). Is this the functional derivative now or just delta S divided by delta gamma ?

Thanks!

Leb said:
Not really a homework question, just the notes are confusing me.

## Homework Statement

Let S be a functional.

(Given without proof)
If S is differentiable its derivative $\delta S$ is uniquely defined as
$\delta S = \int_{x_{0}}^{x_{1}}\frac{\delta S}{\delta \gamma} \delta \gamma dx$ where $\frac{\delta S}{\delta \gamma}$

For functional S
$S[\gamma]=\int_{x_{0}}^{x_{1}}L(x,y(x),y'(x))dx$
the variation is defined by
$\delta S=\int_{x_{0}}^{x_{1}}\left( \frac{\partial{L}}{\partial{y}}-\frac{d}{dx}\frac{\partial{L}}{\partial{y'}}\right)\delta \gamma dx + \left.\frac{\partial{L}}{\partial{y'}}\delta \gamma \right|_{x_{0}}^{x_{1}}$
If $h(x_0)= h(x_1)= 0$, that last term is 0 and the result follows by differentiating both sides.

Define $\delta \gamma = \epsilon h(x)$ where $\epsilon =const.<< 1$ and h(x) is an arbitrary (perturbation) function.

Now, the notes given by the lecturer say, that if end points are fixed i.e. $h(x_{0})=h(x_{1})=0$ the variation simplifies to

$\frac{\delta S}{\delta \gamma}= \frac{\partial{L}}{\partial{y}}-\frac{d}{dx}\frac{\partial{L}}{\partial{y'}}$

I do not understand, how do we get this (I get how the last term in the variation vanishes). Is this the functional derivative now or just delta S divided by delta gamma ?

Thanks!

Thanks for your input.I get how the last term vanishes. But how do you differentiate the first term ? How does $\frac{\delta S}{\delta \gamma}$ appear ?

Last edited:

## 1. What is "Variation of S with fixed end points"?

"Variation of S with fixed end points" refers to the study of how a particular quantity or variable, denoted as S, changes or varies when the endpoints or boundaries are held constant or fixed. This can be applied to various fields such as mathematics, physics, and economics.

## 2. Why is it important to study the variation of S with fixed end points?

Studying the variation of S with fixed end points can provide valuable insights into the behavior and relationships of different variables. It can also help in making predictions and understanding the fundamental principles governing various phenomena in different fields of study.

## 3. What are some real-life examples of "Variation of S with fixed end points"?

One example is the study of elasticity in economics, where the price of a product (S) is observed to change with the quantity demanded and supplied, while the endpoints (consumer income and production capacity) remain fixed. Another example is the study of the displacement of an object (S) with respect to time in physics, when the initial and final positions of the object are fixed.

## 4. How is the variation of S with fixed end points represented mathematically?

The variation of S with fixed end points can be represented using a function or equation, where the variable S is dependent on other variables and parameters that are held constant. For example, S = f(x), where x is a fixed variable and f(x) is a function describing the variation of S.

## 5. What are some common techniques used to analyze the variation of S with fixed end points?

Some common techniques include graphical analysis, where the variation of S is plotted against the fixed end points to observe patterns and relationships. Another technique is calculus, which can be used to find the rate of change of S with respect to the fixed end points. Statistical methods can also be applied to analyze the variation of S in different data sets.

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