✎ Hamilton-Jacobi Equation [The Physics Travel Guide]

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====== Hamilton-Jacobi Equation ======

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Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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For a very nice explanation of the origin of the Hamilton-Jacobi equation and its meaning see chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan

<blockquote>The Hamilton-Jacobi equation is essentially a dispersion relation for
a complex wave. This is easy to see in the context of non-relativistic
[[theories:quantum_mechanics|quantum mechanics]]. If a quantum amplitude is expressed in the
form ψ = Rexp(iS/h¯), then the Hamilton-Jacobi equation relates
p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V).
This is a relation between the wave vector k = p/h¯ and the frequency
ω = E/h¯ of the “matter wave” associated with the particle.<cite>chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan
</cite></blockquote>

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Take note that while the Hamilton equation for time-evolution is closely related to the Heisenberg equation in quantum mechanics, the Hamilton-Jacobi equation is closely connected to the [[equations:schroedinger_equation|Schrödinger equation]]. See, for example, Section 4.8.1 in [[http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf|Tong's lecture notes]].

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
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