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

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SUMMARY

The discussion clarifies the formal difference between Δ-variation and δ-variation in the context of calculus of variations. Specifically, Δ-variation is utilized to derive the principle of least action, while δ-variation is employed within the calculus of variations framework. The relationship Δ = δ + Δt*(d/dt) is established, indicating that Δ-variation encompasses both δ-variation and a temporal component. Understanding these distinctions is crucial for accurately applying these concepts in mathematical physics and variational calculus.

PREREQUISITES
  • Understanding of calculus of variations
  • Familiarity with the principle of least action
  • Knowledge of differential calculus
  • Basic grasp of mathematical notations and symbols
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  • Study the derivation of the principle of least action in classical mechanics
  • Explore the applications of δ-variation in variational calculus
  • Investigate the implications of inexact differentials in mathematical analysis
  • Review advanced texts on calculus of variations for deeper insights
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Students and professionals in mathematics, physicists, and anyone studying advanced calculus concepts, particularly those focusing on variational methods and their applications in physics.

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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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Do you have any specific references for these notations?
 

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