Find S, Hamilton Principal Function from HJE

[bluejay27]

How do you find the Hamilton principal function, S? From the Hamilton Jacobi equation if it is not given.

 

[jedishrfu]

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.

 

[bluejay27]

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?