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bluejay27

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In summary, the Hamilton-Jacobi equation (HJE) can be used to find the principal function S, which is considered the most useful solution. This can be achieved through separation of variables, as mentioned in discussions of the HJE. Examples and more detailed treatments can be found in various sources. While there is a method for finding S in the time independent case, there is not a direct method for the time dependent case.

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bluejay27

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jedishrfu

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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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bluejay27

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I like the fact that they use for the time independent treatment for S. Is there one for the time dependent one?jedishrfu said:

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.

The Hamilton-Jacobi Equation is a partial differential equation that describes the dynamics of a classical system in terms of a single function known as the Hamilton Principal Function. It is closely related to Hamilton's equations of motion and is often used in the study of classical mechanics and quantum mechanics.

The Hamilton Principal Function is a powerful tool in classical mechanics as it can be used to solve for the equations of motion of a system without having to explicitly solve for the equations of motion themselves. This makes it a useful technique for systems with complex dynamics and can provide insights into the behavior of the system.

The Hamilton Principal Function can be found by solving the HJE, which involves finding a function that satisfies the partial differential equation. This can be done using various mathematical techniques such as separation of variables or the method of characteristics.

The HJE and Hamilton Principal Function are applicable to any classical system that can be described by a Hamiltonian, which is a function that represents the total energy of the system. This includes many physical systems, such as particles in a gravitational field or a simple pendulum.

The HJE and Hamilton Principal Function have many practical applications in physics, engineering, and other fields. They are used in the study of celestial mechanics, quantum mechanics, and control theory. They are also useful in understanding the behavior of complex systems, such as weather patterns and financial markets.

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