Is the Hamilton-Jacobi Equation Enough for Hamiltonian Dynamics?

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The Hamilton-Jacobi equation (HJE) serves as a necessary condition for extremal geometry in calculus of variations and is related to the Hamiltonian framework in classical mechanics. While it is a crucial part of Hamiltonian mechanics, it is not strictly equivalent to Hamilton's equations, which describe the evolution of a system. The HJE can be seen as a special case of the Hamilton-Jacobi-Bellman equation, emphasizing its role in optimization problems. Some sources suggest that the HJE provides insights into the dynamics described by Hamilton's equations but does not fully encapsulate them. Overall, the distinction between the HJE and Hamilton's equations is important for understanding their respective roles in physics and mathematics.
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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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