Peeter Joot's (OLD) Blog.

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Posts Tagged ‘non-homogeneous first order linear differential equation’

Desai Chapter 10 notes.

Posted by peeterjoot on December 7, 2010

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Motivation.

Chapter 10 notes for [1].

Notes

In 10.3 (interaction with a electric field), Green’s functions are introduced to solve the first order differential equation

\begin{aligned}\frac{da}{dt} + i \omega_0 a = - i \omega_0 \lambda(t)\end{aligned} \hspace{\stretch{1}}(2.1)

A simpler way is to use the usual trick of assuming that we can take the constant term in the homogeneous solution and allow it to vary with time.

Since our homogeneous solution is of the form

\begin{aligned}a_H(t) = a_H(0) e^{-i\omega_0 t},\end{aligned} \hspace{\stretch{1}}(2.2)

we can look for a specific solution to the forcing term equation of the form

\begin{aligned}a_S(t) = f(t) e^{-i\omega_0 t}\end{aligned} \hspace{\stretch{1}}(2.3)

We get

\begin{aligned}f' = -i \omega_0 \lambda(t) e^{i \omega_0 t}\end{aligned} \hspace{\stretch{1}}(2.4)

which can be integrate directly to find the non-homogeneous solution

\begin{aligned}a_S(t) = a_S(t_0) e^{-i \omega_0 (t - t_0)} - i \omega_0 \int_{t_0}^t \lambda(t') e^{-i \omega_0 (t-t')} dt'\end{aligned} \hspace{\stretch{1}}(2.5)

Setting t_0 = -\infty, with a requirement that a_S(-\infty) = 0 and adding in a general homogeneous solution one then has 10.92 without the complications of Green’s functions or the associated contour integrals. I suppose the author wanted to introduce this as a general purpose tool and this was a simple way to do so.

His introduction of Green’s functions this way I didn’t personally find very clear. Specifically, he doesn’t actually define what a Green’s function is, and the Appendix 20.13 he refers to only discusses the subtlies of the associated Contour integration. I didn’t understand where equation 10.83 came from in the first place.

Something like the following would have been helpful (the type of argument found in [2])

Given a linear operator L, such that L u(x) = f(x), we search for the Green’s function G(x,s) such that L G(x,s) = \delta(x-s). For such a function we have

\begin{aligned}\int L G(x,s) f(s) ds &= \int \delta(x-s) f(s) ds \\ &= f(x)\end{aligned}

and by linearity we also have

\begin{aligned}f(x) &=\int L G(x,s) f(s) ds \\ &= L \int G(x,s) f(s) ds \\ \end{aligned}

and can therefore identify u(x) = \int G(x,s) f(s) ds as the desired solution to L u(x) = f(x) once the Green’s function G(x,s) associated with operator L has been determined.

References

[1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.

[2] Wikipedia. Green’s function — wikipedia, the free encyclopedia [online]. 2010. [Online; accessed 20-November-2010]. http://en.wikipedia.org/w/index.php?title=Green%27s_function&oldid=3911%86019.

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Hamiltonian treatment of the rigid pendulum problem.

Posted by peeterjoot on October 7, 2009

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The bob speed for a stiff rod of length l is (l \dot{\theta})^2, and our potential is m g h = m g l (1 -\cos\theta). The Lagrangian is therefore

\begin{aligned}\mathcal{L} = \frac{1}{{2}} m l^2 \dot{\theta}^2 - mg l (1 -\cos\theta) \\ \end{aligned} \quad\quad\quad(121)

The constant m g l term can be dropped, and our canonical momentum conjugate to \dot{\theta} is p_\theta = m l^2 \dot{\theta}, so our Hamiltonian is

\begin{aligned}H = \frac{1}{{2 m l^2}} {p_\theta}^2 - mg l \cos\theta \\ \end{aligned} \quad\quad\quad(123)

We can now compute the Hamiltonian equations

\begin{aligned}\frac{\partial {H}}{\partial {p_\theta}}  &= \dot{\theta}         = \frac{1}{{ m l^2 }} p_\theta \\ \frac{\partial {H}}{\partial {q}}         &= -\dot{p}_\theta   = m g l \sin\theta\end{aligned} \quad\quad\quad(125)

Only in the neighborhood of a particular angle can we write this in matrix form. Suppose we expand this around \theta = \theta_0 + \alpha. The sine is then

\begin{aligned}\sin\theta \approx \sin\theta_0 + \cos\theta_0 \alpha\end{aligned} \quad\quad\quad(127)

The linear approximation of the Hamiltonian equations after a change of variables become

\begin{aligned}\frac{d}{dt}\begin{bmatrix}p_\theta \\ \alpha\end{bmatrix}=\begin{bmatrix}0 & -m g l \cos\theta_0 \\ 1/ m l^2 & 0\end{bmatrix}\begin{bmatrix}p_\theta \\ \alpha\end{bmatrix}-m g l \sin\theta_0\begin{bmatrix}1 \\ 0\end{bmatrix}\end{aligned} \quad\quad\quad(128)

A change of variables that scales the factors in the matrix to have equal magnitude and equivalent dimensions is helpful. Writing

\begin{aligned}\begin{bmatrix}p_\theta \\ \alpha\end{bmatrix}=\begin{bmatrix}a & 0 \\ 0 & 1\end{bmatrix}\mathbf{z}\end{aligned} \quad\quad\quad(129)

one finds

\begin{aligned}\frac{d\mathbf{z}}{dt}&=\begin{bmatrix}0 & -m g l \cos\theta_0/a \\ a/ m l^2 & 0\end{bmatrix}\mathbf{z}-\frac{m g l \sin\theta_0 }{a}\begin{bmatrix}1 \\ 0\end{bmatrix} \\ \end{aligned} \quad\quad\quad(130)

To tidy this up, we want

\begin{aligned}{\left\lvert{\frac{a}{m l^2}}\right\rvert} = {\left\lvert{\frac{m g l \cos\theta_0}{a}}\right\rvert}\end{aligned} \quad\quad\quad(132)

Or

\begin{aligned}a = m l^2 \sqrt{\frac{g}{l} {\left\lvert{\cos\theta_0}\right\rvert}}\end{aligned} \quad\quad\quad(133)

The result of applying this scaling is quite different above and below the horizontal due to the sign difference in the cosine. Below the horizontal where \theta_0 \in (-\pi/2, \pi/2) we get

\begin{aligned}\frac{d\mathbf{z}}{dt}&=\sqrt{\frac{g}{l} \cos\theta_0}\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\mathbf{z}-\sin\theta_0 \sqrt{\frac{g}{l \cos\theta_0}}\begin{bmatrix}1 \\ 0\end{bmatrix}\end{aligned} \quad\quad\quad(134)

and above the horizontal where \theta_0 \in (\pi/2, 3\pi/2) we get

\begin{aligned}\frac{d\mathbf{z}}{dt}&=\sqrt{-\frac{g}{l} \cos\theta_0}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\mathbf{z}-\sin\theta_0 \sqrt{-\frac{g}{l \cos\theta_0}}\begin{bmatrix}1 \\ 0\end{bmatrix}\end{aligned} \quad\quad\quad(135)

Since (\begin{smallmatrix} 0 & -1 \\  1 & 0 \end{smallmatrix}) has the characteristics of an imaginary number (squaring to the negative of the identity) the homogeneous part of the solution for the change of the phase space vector in the vicinity of any initial angle in the lower half plane is trigonometric. Similarly the solutions are necessarily hyperbolic in the upper half plane since (\begin{smallmatrix} 0 & 1 \\  1 & 0 \end{smallmatrix}) squares to identity. And around \pm \pi/2 something totally different (return to this later). The problem is now reduced to solving a non-homogeneous first order matrix equation of the form

\begin{aligned}\mathbf{z}' = \Omega \mathbf{z} + \mathbf{b}\end{aligned} \quad\quad\quad(136)

But we have the good fortune of being able to easily exponentiate and invert this matrix \Omega. The homogeneous problem

\begin{aligned}\mathbf{z}' = \Omega \mathbf{z}\end{aligned} \quad\quad\quad(137)

has the solution

\begin{aligned}\mathbf{z}_h(t) = e^{\Omega t} \mathbf{z}_{t=0}\end{aligned} \quad\quad\quad(138)

Assuming a specific solution z = e^{\Omega t}f(t) for the non-homogeneous equation, one finds z = \Omega^{-1} (e^{\Omega t} - I) \mathbf{b}. The complete solution with both the homogeneous and non-homogeneous parts is thus

\begin{aligned}\mathbf{z}(t) = e^{\Omega t} \mathbf{z}_0 + \Omega^{-1} (e^{\Omega t} - I) \mathbf{b}\end{aligned} \quad\quad\quad(139)

Going back to the pendulum problem, lets write

\begin{aligned}\omega = \sqrt{\frac{g}{l} {\left\lvert{\cos\theta_0}\right\rvert}}\end{aligned} \quad\quad\quad(140)

Below the horizontal we have

\begin{aligned}\Omega&= \omega\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix} \\ \Omega^{-1}&= -\frac{1}{{\omega}} \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix} \\ e^{\Omega t}&=\cos(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sin(\omega t)\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\end{aligned} \quad\quad\quad(141)

Whereas above the horizontal we have

\begin{aligned}\Omega&= \omega\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \\ \Omega^{-1}&= \frac{1}{{\omega}} \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \\ e^{\Omega t}&=\cosh(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sinh(\omega t)\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\end{aligned} \quad\quad\quad(144)

In both cases we have

\begin{aligned}\begin{bmatrix}p_\theta \\ \alpha\end{bmatrix}&=\begin{bmatrix}m l^2 \omega & 0 \\ 0 & 1\end{bmatrix}\mathbf{z} \\ \mathbf{b} &= - \frac{g}{l}\frac{\sin\theta_0}{\omega}\begin{bmatrix}1 \\ 0\end{bmatrix}\end{aligned} \quad\quad\quad(147)

(where the real angle was \theta = \theta_0 + \alpha). Since in this case \Omega^{-1} and e^{\Omega t} commute, we have below the horizontal

\begin{aligned}\mathbf{z}(t)&= e^{\Omega t} (\mathbf{z}_0 - \Omega^{-1} \mathbf{b}) - \Omega^{-1} \mathbf{b} \\ &=\left(\cos(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sin(\omega t)\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\right)\left(\mathbf{z}_0 +\frac{1}{{\omega}} \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\mathbf{b} \right)+\frac{1}{{\omega}} \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\mathbf{b} \end{aligned}

Expanding out the \mathbf{b} terms and doing the same for above the horizontal we have respectively (below and above)

\begin{aligned}\mathbf{z}_\text{low}(t)&=\left(\cos(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sin(\omega t)\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\right)\left(\mathbf{z}_0 -\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\right)-\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix} \\ \mathbf{z}_\text{high}(t)&=\left(\cosh(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sinh(\omega t)\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\right)\left(\mathbf{z}_0 +\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\right)+\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\end{aligned} \quad\quad\quad(149)

The only thing that is really left is re-insertion of the original momentum and position variables using the inverse relation

\begin{aligned}\mathbf{z} &=\begin{bmatrix}1/(m l^2 \omega) & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}p_\theta \\ \theta - \theta_0\end{bmatrix}\end{aligned} \quad\quad\quad(151)

Will that final insertion do anything more than make things messier? Observe that the \mathbf{z}_0 only has a momentum component when expressed back in terms of the total angle \theta. Also recall that p_\theta = m l^2 \dot{\theta}, so we have

\begin{aligned}\mathbf{z} &=\begin{bmatrix}\dot{\theta}/\omega \\ \theta - \theta_0\end{bmatrix} \\ \mathbf{z}_0&=\begin{bmatrix}\dot{\theta}_{t=0}/\omega \\ 0\end{bmatrix} \\ \end{aligned} \quad\quad\quad(152)

If this is somehow mystically free of all math mistakes then we have the final solution

\begin{aligned}{\begin{bmatrix}\dot{\theta}(t)/\omega \\ \theta(t) - \theta_0\end{bmatrix}}_\text{low}&=\left(\cos(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sin(\omega t)\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\right)\left(\frac{\dot{\theta}(0)}{\omega}\begin{bmatrix}1 \\ 0\end{bmatrix}-\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\right)-\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix} \\ {\begin{bmatrix}\dot{\theta}(t)/\omega \\ \theta(t) - \theta_0\end{bmatrix}}_\text{high}&=\left(\cosh(\omega t)\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\sinh(\omega t)\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\right)\left(\frac{\dot{\theta}(0)}{\omega}\begin{bmatrix}1 \\ 0\end{bmatrix}+\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\right)+\frac{g}{l}\frac{\sin\theta_0}{\omega^2} \begin{bmatrix}0 \\ 1 \end{bmatrix}\end{aligned} \quad\quad\quad(155)

A qualification is required to call this a solution since it is only a solution is the restricted range where \theta is close enough to \theta_0 (in some imprecisely specified sense). One could conceivably apply this in a recursive fashion however, calculating the result for a small incremental change, yielding the new phase space point, and repeating at the new angle.

The question of what the form of the solution in the neighborhood of \pm \pi/2 has also been ignored. That’s probably also worth considering but I don’t feel like trying now.

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