Calculus of Variations: Δ-Variation vs. δ- Variation

In summary, the Δ-variation and the δ-variation are often used interchangeably, but they have different meanings in the context of calculus of variations and the principle of least action. The Δ is used to represent a change in some quantity, while the δ corresponds to an inexact differential. Δ = δ + Δt*(d/dt), however, it is unclear how this equation is derived. The δ-variation is employed in the calculus of variations, while the Δ is used to derive the principle of least action. Further insight and specific references on these notations are welcome.
  • #1
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Does anybody know what is the formal difference between the Δ -variation and the δ- variation is?

They seem to be used interchangeably. I read somewhere that Δ = δ + Δt*(d/dt) but I have no idea how that is arrived at.

I know that the δ- variation is employed in the calculus of variations and the Δ is used to derive the principle of least action. As I previously understood it, the Δ referred to a "change in" some quantity and the δ corresponds to a inexact differential, but how are they defined in this context of calculus of variations (and the principle of least action)?

Any further insight would be greatly appreciated!
 
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  • #2
Do you have any specific references for these notations?
 

1. What is the difference between Δ-variation and δ-variation in calculus of variations?

The Δ-variation is a finite difference, while the δ-variation is an infinitesimal difference. In other words, Δ-variation involves a finite change in the function, while δ-variation involves an infinitely small change.

2. How are Δ-variation and δ-variation used in calculus of variations?

Δ-variation and δ-variation are used to find the extrema of a functional, which is a function of a function. The extrema are found by setting the δ-variation of the functional equal to zero and solving for the function.

3. Can both Δ-variation and δ-variation be used interchangeably in calculus of variations?

No, they cannot be used interchangeably. The Δ-variation is used for discrete functions, while the δ-variation is used for continuous functions. It is important to use the correct variation depending on the type of function being analyzed.

4. What is the relationship between Δ-variation and δ-variation?

The Δ-variation can be seen as a discrete approximation of the δ-variation. As the step size of the Δ-variation approaches zero, it becomes more and more similar to the δ-variation. In other words, δ-variation is the limit of Δ-variation as the step size approaches zero.

5. How do Δ-variation and δ-variation relate to the Euler-Lagrange equation in calculus of variations?

The Euler-Lagrange equation is derived using the δ-variation of the functional. It represents the necessary condition for a function to be an extremum of the functional. However, in certain cases, using the Δ-variation instead of the δ-variation can also lead to the same Euler-Lagrange equation.

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