The δ Notation in Calculus of Variations

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SUMMARY

The δ notation, introduced in section 6.7 of the 5th edition of "Classical Dynamics of Particles and Systems" by Stephen T. Thornton and Jerry B. Marion, represents variations in calculus of variations. Specifically, the equations $$\delta J = \frac{\partial J}{\partial \alpha}d\alpha$$ and $$\delta y = \frac{\partial y}{\partial \alpha}d\alpha$$ define how variations in the path affect the functional J and the variable y. The right-hand sides of these equations illustrate how small changes in the parameter α lead to variations from the actual path, providing a geometrical interpretation of the δ notation.

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  • Understanding of calculus, particularly differentiation and integration.
  • Familiarity with the concepts of functionals in calculus of variations.
  • Knowledge of the principles of classical mechanics as outlined in "Classical Dynamics of Particles and Systems".
  • Basic comprehension of geometric interpretations in mathematical contexts.
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  • Study the derivation of the Euler-Lagrange equation in calculus of variations.
  • Explore the application of δ notation in solving variational problems.
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  • Review geometric interpretations of calculus concepts through visual aids and diagrams.
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Students of physics, mathematicians, and researchers interested in the calculus of variations and its applications in classical mechanics and optimization problems.

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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.
 
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Maybe this image I have found on Wikipedia helps a bit:
1280px-Totales_Differential.png


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