Back to the on-screen lesson ·
Phase portraits, centers, saddles and separatrices, the oscillator's phase ellipse, Liouville's theorem of conserved phase-space volume, damping, attractors and emittance.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to draw and read phase portraits, classify fixed points, and use Liouville's theorem to relate spreads in position and momentum.
The last lesson wrote mechanics as Hamilton's first-order equations in coordinates and momenta, and drew motions as contours of $\mathcal{H}$ in the phase plane. You know the divergence of a vector field and the divergence theorem. This lesson studies the phase-space flow as a whole: its fixed points, the curves that divide one kind of motion from another, and the theorem that the flow preserves volume.
| Term | What it means |
|---|---|
| Phase space | The space of all $(q, p)$; one point is one complete state of the system. |
| Phase flow | The motion of every point of phase space under Hamilton's equations. |
| Fixed point | A state where $\dot{q} = \dot{p} = 0$: an equilibrium. |
| Center | A fixed point surrounded by closed orbits, at a minimum of the potential. |
| Saddle | A fixed point where orbits approach along one direction and leave along another. |
| Separatrix | An orbit through a saddle, dividing regions of different kinds of motion. |
| Liouville's theorem | Hamiltonian flow conserves phase-space volume. |
| Emittance | The phase-space area of a particle beam, conserved as the beam is steered and focused. |
A system with $n$ degrees of freedom has a $2n$-dimensional phase space. Hamilton's equations give every point a velocity, $\vec{v} = (\partial\mathcal{H}/\partial p, -\partial\mathcal{H}/\partial q)$, so the whole space flows. Where that velocity vanishes are the fixed points. For $\mathcal{H} = p^2/2m + U(q)$ they sit at $p = 0$ and $U'(q) = 0$: at a minimum of $U$ the orbits around them are closed, a center; at a maximum they split, a saddle. The orbits through a saddle, the separatrices, divide the phase space into regions of qualitatively different motion.
The divergence of the phase flow is
$$\frac{\partial\dot{q}}{\partial q} + \frac{\partial\dot{p}}{\partial p} = \frac{\partial^2\mathcal{H}}{\partial q\,\partial p} - \frac{\partial^2\mathcal{H}}{\partial p\,\partial q} = 0.$$
A flow with zero divergence neither compresses nor expands: a region of phase space keeps its volume as it moves, however distorted its shape becomes. This is Liouville's theorem. With friction the divergence is negative, volumes shrink, and states can collect on an attractor; without friction they cannot.
Another way: picture
Put a drop of ink into a stream of water that neither compresses nor expands. The drop stretches into a thin thread, twists around rocks and folds back on itself, but its volume never changes. Phase space under Hamiltonian motion is that stream, and a cloud of possible states is the ink: it can become a long, fine filament, but it cannot be squeezed into a point.
Another way: steps
For a harmonic oscillator, $\mathcal{H} = p^2/2m + \tfrac{1}{2}m\omega^2x^2 = E$ is an ellipse with semi-axes $A = \sqrt{2E/m\omega^2}$ along $x$ and $m\omega A$ along $p$. The state goes round it clockwise once per period. Its area is $\pi \cdot A \cdot m\omega A = 2\pi E/\omega$.
That area, the action, has units of joule-seconds. In 1900 Max Planck found that the energies of oscillators in radiation came in steps of $\hbar\omega$, which is the same as saying that the area of the phase ellipse comes in steps of $h = 2\pi\hbar$. The first rule of quantum physics was a rule about areas in phase space, and Liouville's theorem is why that rule could be stated in a way that does not depend on the coordinates.
Near a fixed point, expand $\mathcal{H}$ to second order. At a minimum of $U$, the contours are small ellipses: a center, where nearby states oscillate. At a maximum, $\mathcal{H} \approx p^2/2m - \tfrac{1}{2}|U''|x^2$ and the contours are hyperbolas: a saddle, with two special directions, one along which states approach and one along which they leave, exponentially.
For a pendulum, the bottom is a center and the inverted position a saddle, with energy $2mgL$. The separatrix at that energy is the orbit that takes infinitely long to creep up to the top. Inside it, the pendulum swings; outside it, it whirls around the pivot. Every Hamiltonian system's phase portrait is organized by its fixed points and the separatrices that join them.
Take a small rectangle of states, $\delta q$ by $\delta p$. In a short time $\delta t$ its left edge moves to $q + \dot{q}\delta t$ and its right edge to $q + \delta q + (\dot{q} + \partial\dot{q}/\partial q\,\delta q)\delta t$, so its width grows by the factor $1 + (\partial\dot{q}/\partial q)\delta t$. Its height grows by $1 + (\partial\dot{p}/\partial p)\delta t$. The area changes by the factor $1 + (\partial\dot{q}/\partial q + \partial\dot{p}/\partial p)\delta t$, and the bracket is zero.
Any region is a union of such rectangles, so every region keeps its area, and in higher dimensions its volume. The theorem needs nothing about the particular Hamiltonian; it follows from the structure of Hamilton's equations alone, the minus sign in the second equation being exactly what makes the two stretching factors cancel.
Add damping to an oscillator: $\dot{p} = -kx - (b/m)p$. The divergence of the flow is now $-b/m$, so phase-space areas shrink as $e^{-bt/m} = e^{-2\beta t}$. Every starting state spirals in to the single point of rest, which is an attractor: a set that states approach and never leave.
Attractors are impossible in Hamiltonian systems, because a region of states can never shrink onto a point. That is the dividing line between conservative and dissipative mechanics. Driven damped systems can have strange attractors, fractal sets of zero volume on which motion is chaotic, the subject of the last lesson. A frictionless system can be chaotic too, but its chaos stretches and folds without ever shrinking.
Liouville's theorem limits what can be done to a beam of particles. Magnets steer and focus a beam by conservative forces, so the beam's phase-space area, its emittance, cannot be reduced by them: squeezing the beam's width spreads its angles. To make a denser beam, accelerator physicists must use dissipation, as in the stochastic cooling developed for antiproton beams, which won a Nobel Prize and was used at Fermilab's Tevatron.
The same theorem underlies statistical mechanics: because phase-space volume is conserved, equal volumes of phase space can be given equal probability. It also raises a puzzle. If volume is conserved, how does entropy grow? The answer is that a compact region is stretched into fine filaments that fill the available space; its volume is unchanged, but any coarse-grained description sees it spread out.
Checking an answer. The divergence of a Hamiltonian flow must come out zero; if it does not, a sign is wrong. Orbits must move right in the upper half plane for $\mathcal{H} = p^2/2m + U$. Centers must sit at minima of $U$ and saddles at maxima. And a conserved-area argument must never produce a spread that is smaller in both $q$ and $p$ at once.
Liouville's theorem has a surprising consequence. If a system's motion is confined to a bounded region of phase space, as for a gas in a box with fixed energy, then almost every state eventually returns arbitrarily close to where it started. Henri Poincaré proved this in 1890: a volume-preserving flow in a finite space cannot keep sending a region into fresh territory forever, because it would run out of room.
For a gas of many molecules the recurrence time is unimaginably long, far longer than the age of the universe, so the gas never visibly returns to its starting arrangement. But for systems with few degrees of freedom, recurrences are seen, and they are one of the tools physicists use to tell regular motion from chaos in numerical simulations.
With two degrees of freedom phase space has four dimensions, and a conserved energy confines each motion to a three-dimensional surface within it. If there is a second conserved quantity, as angular momentum is for a central force, the motion is confined further, to a two-dimensional surface shaped like the skin of a doughnut, a torus. Motion on a torus is regular: two angles advancing at steady rates, like the radial and angular oscillations of an orbit.
Systems with as many independent conserved quantities as degrees of freedom are called integrable, and every system solved in this course so far is one. Most systems are not. The double pendulum and three gravitating bodies have too few conserved quantities, their states can wander over the whole energy surface, and their motion can be chaotic. Liouville's theorem still holds for them, which is what makes the statistical description of such motion possible.
At Fermilab, west of Chicago, proton beams are steered and focused by magnets on their way to targets and detectors. Each proton's transverse state is a position and an angle, a point in phase space, and the beam is a cloud of such points. Its area, the emittance, is fixed by Liouville's theorem as long as the forces are conservative.
So an accelerator physicist who wants a tighter spot must accept a larger angular spread: a beam $12$ mm wide with $2$ milliradians of spread, focused to $4$ mm, comes out with $6$ milliradians. For experiments that need both a small spot and a small spread, the only way out is to reduce the emittance by dissipation, which is what electron cooling and stochastic cooling do, and why cooling systems were among the key technologies of the Tevatron's antiproton source.
The atmosphere is not Hamiltonian, since friction and heating matter, but over short times its large-scale motion behaves much like a conservative system. Forecasters at the National Weather Service represent the uncertainty in today's conditions by running an ensemble of forecasts, a cloud of starting states in the model's enormous phase space.
As the forecast runs, the cloud stretches along some directions and squeezes along others, much as Liouville's theorem describes, and after a week or two it has spread across a large fraction of the possible weather states. At that point the ensemble no longer predicts anything definite. The ensemble's spread, drawn as the fan of possible hurricane tracks on the evening news, is a direct picture of a phase-space region being stretched by the flow.
It is natural to picture any system settling down to rest or to a steady cycle. Settling means many starting states end up near one final state, which squeezes a region of phase space down onto a point or a curve. Liouville's theorem forbids that for Hamiltonian systems: phase-space volume is conserved. Only friction or other dissipation, which makes the divergence negative, allows attractors.
A second error is to think Liouville's theorem keeps a region's shape. It keeps only its volume; the shape typically shears and stretches, sometimes into a long thin thread that winds through the whole accessible space.
Write the energy of a harmonic oscillator.
$\dfrac{p^2}{2m} + \tfrac{1}{2}m\omega^2x^2 = E$
A contour of $\mathcal{H}$.
Put it in the standard form of an ellipse.
$\dfrac{x^2}{2E/m\omega^2} + \dfrac{p^2}{2mE} = 1$
Divide by $E$.
Read off the semi-axes.
$a_x = \sqrt{\dfrac{2E}{m\omega^2}}, \quad a_p = \sqrt{2mE}$
Square roots of the denominators.
Multiply for the area.
$\text{area} = \pi a_xa_p = \pi\sqrt{\dfrac{4E^2}{\omega^2}}$
Area of an ellipse.
Simplify the area.
$\text{area} = \dfrac{2\pi E}{\omega}$
Proportional to energy over frequency.
Write the phase velocity.
$\vec{v} = (\dot{q}, \dot{p}) = \left(\dfrac{\partial\mathcal{H}}{\partial p}, -\dfrac{\partial\mathcal{H}}{\partial q}\right)$
Hamilton's equations.
Differentiate the first component with respect to position.
$\dfrac{\partial\dot{q}}{\partial q} = \dfrac{\partial^2\mathcal{H}}{\partial q\,\partial p}$
Stretching along $q$.
Differentiate the second with respect to momentum.
$\dfrac{\partial\dot{p}}{\partial p} = -\dfrac{\partial^2\mathcal{H}}{\partial p\,\partial q}$
Stretching along $p$.
Add the two derivatives.
$\nabla \cdot \vec{v} = 0$
Mixed partials are equal.
Apply the divergence theorem to a region.
$\dfrac{dA}{dt} = \displaystyle\oint\vec{v} \cdot \hat{n}\,ds = \int\nabla \cdot \vec{v}\,dA = 0$
The rate of change of area is the net outflow through the boundary.
Write the damped oscillator's phase equations.
$\dot{x} = \dfrac{p}{m}, \quad \dot{p} = -kx - \dfrac{b}{m}p$
Newton's law with a drag term.
Compute the divergence.
$\dfrac{\partial\dot{x}}{\partial x} + \dfrac{\partial\dot{p}}{\partial p} = 0 - \dfrac{b}{m}$
Only the drag depends on $p$.
Write the rate of change of area.
$\dfrac{dA}{dt} = -\dfrac{b}{m}A$
The divergence is uniform.
Solve for the area.
$A = A_0e^{-bt/m} = A_0e^{-2\beta t}$
Exponential shrinking at twice the amplitude decay rate.
Find the time to halve the area with $\beta = 0.5$ s⁻¹.
$t = \dfrac{\ln 2}{2\beta} = \dfrac{0.693}{1.0} = 0.69\ \text{s}$
Set $e^{-2\beta t} = \tfrac{1}{2}$.
Interpret the limit.
$A \to 0 \text{ as } t \to \infty$
All states collapse onto the rest point, an attractor.
Find the emittance.
$\varepsilon = 6 \times 2 = 12\ \text{mm mrad}$
Width times angular spread.
Divide by the new width.
$\sigma_\theta' = \dfrac{12}{3}$
The emittance is conserved.
Evaluate the spread.
A small patch of phase space of area $6$ units holds the possible starting states of a frictionless pendulum. As each state evolves by Hamilton's equations, what happens to the area of the patch?
Complete the worked solution: a pendulum has a bob of mass $2$ kg on a light rod of length $1$ m, with $g = 10$ m/s² and energy measured from the bottom. Find the energy of the inverted fixed point in J, $\omega^2$ for small swings in s⁻², and the energy of the bob held at rest with the rod horizontal, in J.
Evaluate the potential at the top.
$U(\pi) = mgL(1 - \cos\pi) = 2mgL =$ s
The saddle, where the separatrix passes.
Divide the gravitational field by the length.
$\omega^2 = \dfrac{g}{L} =$ w
The curvature of the center at the bottom.
Evaluate the potential with the rod horizontal.
$U\left(\tfrac{\pi}{2}\right) = mgL(1 - 0) =$ h
At rest, so all the energy is potential.
Place the horizontal release on the phase portrait.
$\text{below the separatrix energy} \Rightarrow \text{a closed orbit}$
Half the saddle energy, so the pendulum swings back and forth.
Match each feature of a phase portrait to its description.
| circled by closed orbits | orbits arriving and leaving | the orbit through a saddle | volume conserved | |
|---|---|---|---|---|
| center | ||||
| saddle | ||||
| separatrix | ||||
| Liouville's theorem |
A particle has $\mathcal{H} = \dfrac{p^2}{2} + x^3 - 12x$. Its fixed points are at $p = 0$, $x = \pm 2$. Fill in $U''$ at $x = 2$, $U''$ at $x = -2$, and the height of the barrier $U(-2) - U(2)$.
| value | |
|---|---|
| $U''$ at the right fixed point | |
| $U''$ at the left fixed point | |
| barrier height |
A damped oscillator obeys $\dot{x} = p/m$ and $\dot{p} = -kx - (b/m)p$ with $b/m = 3$ s⁻¹. A patch of starting states has phase-space area $5$ units. Write its area as a formula in $t$.
Answer:
A $4$ kg oscillator with $\omega = 4$ rad/s swings with amplitude $4$ m. Its phase-space orbit is an ellipse. What is the ellipse's area divided by $\pi$, in J s?
Answer: J s of phase area over pi
At Fermilab in Illinois, a proton beam has a width of $24$ mm and an angular spread of $4$ milliradians. Magnets focus it to a width of $8$ mm at a target, with no loss of energy or particles. What is its angular spread there, in milliradians?
Answer: mrad of angular spread
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A damped oscillator obeys $\dot{x} = p/m$ and $\dot{p} = -kx - (b/m)p$ with $b/m = 5$ s⁻¹. A patch of starting states has phase-space area $9$ units. Write its area as a formula in $t$.
Answer:
You can use phase space and Liouville's theorem. Explain to someone why magnets alone cannot make a particle beam both narrower and less spread in angle.
19. Your turn: a beam of width $6$ mm and angular spread $2$ mrad is focused to $3$ mm. Find its new angular spread., step 3
$\sigma_\theta' = 4\ \text{mrad}$
Half the width, twice the angle.