A Find S, Hamilton Principal Function from HJE

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To find the Hamilton principal function S from the Hamilton-Jacobi equation (HJE) when it is not provided, separation of variables is a key technique. This method is highlighted as a particularly useful solution for deriving S. The discussion references various resources, including academic papers and textbooks, that elaborate on the application of this technique. Additionally, there is interest in whether a similar approach exists for time-dependent scenarios. Overall, the conversation emphasizes the importance of S in solving the HJE effectively.
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How do you find the Hamilton principal function, S? From the Hamilton Jacobi equation if it is not given.
 
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Here's a discussion of the HJE where it mentions using separation of variables to find the principal function S. It further says the S in this case is considered the most useful solution.

https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation

and here's an example of where the technique was used:

http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMHamiltonJacobi.pdf

and some more detailed treatments:

http://www.physics.rutgers.edu/~shapiro/507/book7_2.pdf

and

https://www.pdx.edu/nanogroup/sites/www.pdx.edu.nanogroup/files/Chapter_4__Hamilton_Variational_principle__Hamilton%20Jacobi_Eq_Classical_Mechanics_1.pdf

Hopefully someone will provide a more direct answer than this.
 
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jedishrfu said:
Here's a discussion of the HJE where it mentions using separation of variables to find the principal function S. It further says the S in this case is considered the most useful solution.

https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation

and here's an example of where the technique was used:

http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMHamiltonJacobi.pdf

and some more detailed treatments:

http://www.physics.rutgers.edu/~shapiro/507/book7_2.pdf

and

https://www.pdx.edu/nanogroup/sites/www.pdx.edu.nanogroup/files/Chapter_4__Hamilton_Variational_principle__Hamilton%20Jacobi_Eq_Classical_Mechanics_1.pdf

Hopefully someone will provide a more direct answer than this.
I like the fact that they use for the time independent treatment for S. Is there one for the time dependent one?
 
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