The δ Notation in Calculus of Variations

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• sams
In summary, the authors of Classical Dynamics of Particles and Systems introduced the δ notation in section 6.7 to represent the variation from the actual path. This is represented by Equations (6.88) which show that the variation in a quantity J or y can be calculated by taking the partial derivative of the quantity with respect to α and multiplying it by the differential dα. The geometrical interpretation of this notation can be understood through the calculus of variations, which deals with finding the path that minimizes or maximizes a certain functional. The image provided on Wikipedia shows a visual representation of this concept.
sams
Gold Member
On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows:

$$\delta J = \frac{\partial J}{\partial \alpha}d\alpha$$ $$\delta y = \frac{\partial y}{\partial \alpha}d\alpha$$

I know that the δδ notation stands for the variation from the actual path, but I cannot relate the geometrical interpretation to the above equation. Can anyone please explain the above terms and provide an explanation on why do the right-hand sides of these relations represent the variation (varied path) from the actual path?

Any help is much appreciated. Thank you so much.

Maybe this image I have found on Wikipedia helps a bit:

(by Muhammet Cakir - Eigenes Werk ; https://de.wikipedia.org/wiki/Totales_Differential)

jedishrfu

1. What is the δ notation in calculus of variations?

The δ notation in calculus of variations is a symbol used to represent a small change or variation in a function. It is typically used in the context of finding the minimum or maximum value of a functional, which is a function that takes in other functions as its inputs.

2. How is the δ notation used in the Euler-Lagrange equation?

In the Euler-Lagrange equation, the δ notation is used to represent a small change in the dependent variable of the functional. This allows us to find the critical points of the functional, which are the points where the first variation (δ) of the functional is equal to zero.

3. Can the δ notation be used in other areas of mathematics?

Yes, the δ notation is not specific to calculus of variations and can be used in other areas of mathematics such as differential equations, optimization problems, and physics. It is a useful tool for representing small changes or variations in a function.

4. What is the relationship between the δ notation and the derivative?

The δ notation and the derivative are closely related, as they both represent the concept of a small change or variation in a function. However, the derivative is used for functions of a single variable, while the δ notation is used for functions of multiple variables or functions of functions. Additionally, the derivative is a limit, while the δ notation is a symbol that represents a small change.

5. Are there any limitations to using the δ notation in calculus of variations?

One limitation of the δ notation is that it can only be used for functions that are differentiable. This means that the function must have a well-defined derivative at every point in its domain. Additionally, the δ notation may not always lead to a solution for a functional, as there may not be a critical point where the first variation is equal to zero.

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