Graduate The δ Notation in Calculus of Variations

[sams]

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.

 

[jedishrfu]

You might find your answer here:

https://en.wikipedia.org/wiki/Calculus_of_variations
and this video sequence that gets into the Calculus of Variations itself:

 

[fresh_42]

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

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