Is the Hamilton-Jacobi Equation Enough for Hamiltonian Dynamics?

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SUMMARY

The Hamilton-Jacobi Equation (HJE) is a necessary condition for Hamiltonian dynamics but is not equivalent to Hamilton's equations. It serves as a formulation of classical mechanics, paralleling other formulations like Newton's laws and Lagrangian mechanics. The HJE is a special case of the Hamilton-Jacobi-Bellman equation, which describes extremal geometry in calculus of variations. The discussion clarifies that while the HJE is integral to understanding Hamiltonian dynamics, it does not replace Hamilton's equations.

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Is Hamilton-Jacobi Equation equivalent to Hamilton equations or it is only a necessary condition ?
Thanks in advance :)
 
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A quick google search finds:
https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation
In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation.[...]In physics, it is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion[citation needed], Lagrangian mechanics and Hamiltonian mechanics.
Does that answer the question? Or am I missing something here?
 
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Sphinx said:
Is Hamilton-Jacobi Equation equivalent to Hamilton equations
I do not think so please look up the following
http://www.srl.caltech.edu/phys106/p106b01/topic2.pdf
 
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