✎ Hamilton-Jacobi Equation [The Physics Travel Guide]

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

<note tip>
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.
</note>
  
<tabbox Concrete>

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>

----

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]].

We consider a manifold $Q$ and a Lagrangian $L \colon TQ\rightarrow\mathbb{R}$. Now, we define Hamilton's principal function
\[
 W \colon Q\times\mathbb{R}\times Q\times\mathbb{R}\longrightarrow\mathbb{R}
\]
by
\[
 W(q_0,t_0; q_1, t_1) = \inf_{q\in\Upsilon} S(q)
\]
where
\[
 \Upsilon=\bigl\{ q \colon [t_0,t_1]\rightarrow Q,\, q(t_0)=q_0,\text{ \& }q(t_1)=q_1 \bigr\}
\]
and
\[
 S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\qdot(t)\Bigr)\,dt .
\]
Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. We have
\[
 \frac{\pa}{\pa q_1^i}W(q_0,q_1) = (p_1)_i, 
\]
\[
\xy
 (48,-3)*+{(q_0,t_0)},(50,0)*\dir{*};(90,10), 
 **\crv{(60,0)&(70,10)&(85,10)&(90,10)},
 (90,7)*+{p_1}, (85,10),(85,10) {\ar@{*\dir{*}->} (90,10)}
 \POS(79.5,7) \ar @{-} (79.5,13)
 \POS(81,7) \ar @{-} (81,13) \POS(82.5,7) \ar @{-} (82.5,13) 
 \POS(84,7) \ar @{-} (84,13) \POS(85.5,7),
 (82.5,5)*+{(q_1,t_1)}
\endxy\]
where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and
\begin{align*}
 \frac{\pa W}{\pa q_0^i}&= -(p_0)_i,\qquad\text{(-momentum at time $t_0$)}\\
 \frac{\pa W}{\pa t_1}&= -H_1,\qquad\text{(-energy at time $t_1$)}\\
 \frac{\pa W}{\pa t_0}&= H_0,\qquad\text{(+energy at time $t_0$)}
\end{align*}
($H_1=H_0$ as energy is conserved).  These last four equations are the Hamilton--Jacobi equations.

</tabbox>

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