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equations:hamilton-jacobi_equation [2017/11/10 16:35] jakobadmin created |
equations:hamilton-jacobi_equation [2018/05/05 12:29] (current) 63.143.42.253 ↷ Links adapted because of a move operation |
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====== Hamilton-Jacobi Equation ====== | ====== Hamilton-Jacobi Equation ====== | ||
- | <tabbox Why is it interesting?> | ||
- | <tabbox Layman> | + | |
+ | <tabbox Intuitive> | ||
<note tip> | <note tip> | ||
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</note> | </note> | ||
| | ||
- | <tabbox Student> | + | <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 | 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 | ||
- | |||
- | <tabbox Researcher> | ||
- | <note tip> | + | <blockquote>The Hamilton-Jacobi equation is essentially a dispersion relation for |
- | The motto in this section is: //the higher the level of abstraction, the better//. | + | a complex wave. This is easy to see in the context of non-relativistic |
- | </note> | + | [[theories:quantum_mechanics:canonical|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> | ||
- | + | ---- | |
- | <tabbox Examples> | + | |
- | --> Example1# | + | 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]]. |
- | |||
- | <-- | ||
- | --> Example2:# | + | <tabbox Abstract> |
- | + | 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 | ||
+ | \[ | ||
+ | S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\dot{q}(t)\Bigr)\,dt . | ||
+ | \] | ||
+ | Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. We have | ||
+ | \[ | ||
+ | \frac{\partial}{\partial q_1^i}W(q_0,q_1) = (p_1)_i, | ||
+ | \] | ||
+ | |||
+ | where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and | ||
+ | $$ \frac{\partial W}{\partial q_0^i}= -(p_0)_i ,\qquad\text{(-momentum at time $t_0$)} $$ | ||
+ | $$ \frac{\partial W}{\partial t_1}= -H_1,\qquad\text{(-energy at time $t_1$)} $$ | ||
+ | $$ \frac{\partial W}{\partial t_0}= H_0,\qquad\text{(+energy at time $t_0$)} | ||
+ | $$ | ||
+ | |||
+ | |||
+ | Take note that $H_1=H_0$ since energy is conserved. These last four equations are the Hamilton--Jacobi equations. | ||
- | <tabbox FAQ> | ||
| | ||
- | <tabbox History> | + | <tabbox Why is it interesting?> |
</tabbox> | </tabbox> | ||