equations:hamilton-jacobi_equation
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equations:hamilton-jacobi_equation [2017/11/10 16:37] jakobadmin [Student] |
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> | ||
| Line 9: | Line 9: | ||
| </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 | ||
| Line 15: | Line 15: | ||
| <blockquote>The Hamilton-Jacobi equation is essentially a dispersion relation for | <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 | a complex wave. This is easy to see in the context of non-relativistic | ||
| - | [[theories:quantum_theory:quantum_mechanics|quantum mechanics]]. If a quantum amplitude is expressed in the | + | [[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 | 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). | p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V). | ||
| Line 21: | Line 21: | ||
| ω = E/h¯ of the “matter wave” associated with the particle.<cite>chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan | 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> | </cite></blockquote> | ||
| - | |||
| - | <tabbox Researcher> | ||
| - | <note tip> | + | ---- |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| - | + | 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]]. | |
| - | <tabbox Examples> | + | |
| - | --> Example1# | ||
| - | + | <tabbox Abstract> | |
| - | <-- | + | |
| - | --> Example2:# | + | 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> | ||
equations/hamilton-jacobi_equation.1510328232.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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