Find S, Hamilton Principal Function from HJE

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SUMMARY

The Hamilton-Jacobi equation (HJE) is a fundamental tool in classical mechanics for finding the Hamilton principal function, S. The separation of variables technique is essential for deriving S when it is not explicitly provided. This method is recognized as the most effective solution for the HJE. For further understanding, resources such as the Wikipedia page on the Hamilton-Jacobi equation and detailed treatments from academic PDFs are invaluable.

PREREQUISITES
  • Understanding of Hamiltonian mechanics
  • Familiarity with the Hamilton-Jacobi equation
  • Knowledge of separation of variables technique
  • Basic concepts of classical mechanics
NEXT STEPS
  • Study the derivation of the Hamilton-Jacobi equation in detail
  • Learn about time-dependent Hamiltonian systems
  • Explore advanced applications of the separation of variables technique
  • Review classical mechanics textbooks focusing on variational principles
USEFUL FOR

This discussion is beneficial for physics students, educators, and researchers focusing on classical mechanics, particularly those interested in the Hamiltonian framework and the Hamilton-Jacobi equation.

bluejay27
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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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