- #1

giraffe714

- 19

- 2

- TL;DR Summary
- I don't understand why the derivative of the Jacobi complete integral with respect to the constant α must be another constant, and furthermore why that constant is negative.

As stated in the TLDR, I don't understand why the derivative of the Jacobi complete integral with respect to the constant α must be another constant, and furthermore why that constant is negative. The textbook I'm following, van Brunt's The Calculus of Variations proves it by taking:

$$ \frac{\partial}{\partial \alpha_1} (H + \frac{\partial S}{\partial t}) = \frac{\partial^2 S}{\partial \alpha_1 \partial t} + \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} = 0 $$

and then just stating that

$$ \frac{\partial S}{\partial \alpha_1} = -\beta_1 $$

is satisfied identically but i cant figure out a) how those two equations are even related and b) from what I can tell, if ## \frac{\partial S}{\partial \alpha_1} = -\beta_1 ## where ##\beta_1## is constant, that means ##\frac{\partial^2 S}{\partial \alpha_1 \partial t} ## must be zero? but if it's zero, then in the original equation

$$ \frac{\partial}{\partial \alpha_1} (H + \frac{\partial S}{\partial t}) = \frac{\partial^2 S}{\partial \alpha_1 \partial t} + \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} = 0 $$

the term ## \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} ## must be zero and I just don't see why that's true?

$$ \frac{\partial}{\partial \alpha_1} (H + \frac{\partial S}{\partial t}) = \frac{\partial^2 S}{\partial \alpha_1 \partial t} + \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} = 0 $$

and then just stating that

$$ \frac{\partial S}{\partial \alpha_1} = -\beta_1 $$

is satisfied identically but i cant figure out a) how those two equations are even related and b) from what I can tell, if ## \frac{\partial S}{\partial \alpha_1} = -\beta_1 ## where ##\beta_1## is constant, that means ##\frac{\partial^2 S}{\partial \alpha_1 \partial t} ## must be zero? but if it's zero, then in the original equation

$$ \frac{\partial}{\partial \alpha_1} (H + \frac{\partial S}{\partial t}) = \frac{\partial^2 S}{\partial \alpha_1 \partial t} + \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} = 0 $$

the term ## \sum_{k=1}^n \frac{\partial^2 S}{\partial \alpha_1 \partial q_k} \frac{\partial H}{\partial p_k} ## must be zero and I just don't see why that's true?