If an ensemble of states occupies a particular volume of phase space
at one moment, then the subsequent evolution of that volume by the
flow described by Hamilton's equations may distort the ensemble but
does not change the volume the ensemble occupies. The fact that phase-space
volume is preserved by the phase flow is called *Liouville's
theorem*.

We will first illustrate the preservation of phase-space volume with a simple example and then prove it in general.

Consider an undriven pendulum described by the Hamiltonian:

In figure 3.25 we see the evolution of an elliptic region around a point on the -axis, in the oscillation region of the pendulum. Three later positions of the region are shown. The region is stretched and sheared by the flow, but the area is preserved. After many cycles, the starting region will be stretched to be a thin layer distributed in the phase angle of the pendulum.

Figure 3.26 shows a similar evolution
(for smaller time intervals) of a region straddling the separatrix^{31}
near the unstable equilibrium point. The phase-space region rapidly
stretches along the separatrix, while preserving the area. The
initial conditions that start in the
oscillation region (inside of the separatrix) will continue to spread
into a thin ring-shaped region, while the initial conditions that
start outside of the separatrix will spread into a thin region of
rotation on the outside of the separatrix.

Consider a set of ordinary differential equations of the form

where *z* is a tuple of *N* state variables.
Let *R*(*t*_{1}) be a region of the state space at time *t*_{1}. Each
element of this region is an initial condition at time *t*_{1} for the
system, and evolves to an element at time *t*_{2} according to the
differential equations. The set of these elements at time *t*_{2} is
the region *R*(*t*_{2}). Regions evolve to regions.

The evolution of the system for a time interval *t* defines a
map *g*_{t, t} from the state space to itself:

Regions map to regions by mapping each element in the region:

The volume *V*(*t*) of a region *R*(*t*) is _{R(t)} , where
is the function whose value is one for every input.
The volume of the evolved region *R*(*t* + *t*) is

where Jac(*g*_{t, t}) is the Jacobian of the
mapping *g*_{t, t}. The Jacobian is the determinant of the
derivative of the mapping.

where *I* is the identity function, so *D**I*(*z*(*t*)) is a unit multiplier.
We can use the fact that if ** A** is an

So the rate of change of the volume at time *t* is

Now we compute *G*_{t} for a system described by a
Hamiltonian *H*.
The components of *z* are the components of the
coordinates and the momenta: *z*^{k} = *q*^{k}, *z*^{k+n} = *p*_{k} for *k* = 0,
`...`, *n* `-` 1. The components of *F* are

for *k* = 0, `...`, *n* `-` 1. The diagonal components of the derivative
_{1} *F* are

The component partial derivatives commute, so the diagonal components
with index *k* and index *k* + *n* are equal and opposite. We see that
the trace, which is the sum of these diagonal components, is zero.
Thus the integral of *G*_{t} over the region *R*(*t*) is zero, so the derivative of
the volume at time *t* is zero. Because *t* is arbitrary, the volume
does not change. This proves *Liouville's theorem*: the phase-space
flow conserves phase-space volume.

Notice that the proof of Liouville's theorem does not depend upon whether or not the Hamiltonian has explicit time dependence. Liouville's theorem holds for systems with time-dependent Hamiltonians.

We may think of the ensemble of all possible states as a fluid flowing around under the control of the dynamics. Liouville's theorem says that this fluid is incompressible for Hamiltonian systems.

**Exercise 3.11.** **Determinants and traces**

Show that equation (3.152) is correct.

Surfaces of section for periodically driven Hamiltonian systems are area preserving if the section coordinates are the phase-space coordinate and momentum. This is an important feature of surfaces of section. It is a consequence of Liouville's theorem for one-degree-of-freedom problems.

It is also the case that surfaces of section such as those we have used for the Hénon-Heiles problem are area preserving, but we are not ready to prove this yet!

The *Poincaré recurrence theorem* is a remarkable theorem that
is a trivial consequence of Liouville's theorem. Loosely, the theorem
states that almost all trajectories eventually return arbitrarily
close to where they started. This is true regardless of whether the
trajectories are chaotic or regular.

More precisely, consider a Hamiltonian dynamical system for which the
phase space is a bounded domain *D*. We identify some initial point
in the phase space, say *z*_{0}. Then, for any finite neighborhood *U*
of *z*_{0} we choose, there are trajectories that emanate from initial
points in that neighborhood and eventually return to the neighborhood.

We can prove this by considering the successive images of *U* under
the time evolution. For simplicity, we restrict consideration to time
evolution for a time interval . The map of the phase space
onto itself generated by time evolution for an interval we
call *C*. Subsequent applications of the map generate a discrete time
evolution. Sets of points in phase space transform by evolving all
the points in the set; the image of the set *U* is denoted *C*(*U*).
Now consider the trajectory of the set *U*, that is, the sets *C*^{n}(*U*)
where *C*^{n} indicates the *n*-times composition of *C*. Now there are
two possibilities: either the successive images *C*^{i}(*U*) intersect or
they do not. If they do not intersect, then with each iteration, a
volume of *D* equal to the volume of *U* gets ``used up'' and cannot
belong to the further image. But the volume of *D* is finite, so we
cannot fit an infinite number of non-intersecting finite volumes into
it. Therefore, after some number of iterations the images intersect.
Suppose *C*^{i}(*U*) intersects with *C*^{j}(*U*), with *j*<*i*, for
definiteness. Then the pre-image of each must also intersect, since
the pre-image of a point in the intersection belongs to both sets.
Thus *C*^{i-1}(*U*) intersects *C*^{j-1}(*U*). This can be continued
until finally we have that *C*^{i-j}(*U*) intersects *U*. So we have proven
that after *i* `-` *j* iterations of the map *C* there are a set of points
initially in *U* that return to the neighborhood *U*.

So for every neighborhood of every point in the phase space there is a subneighborhood such that the trajectories emanating from all of the points in that subneighborhood return to that subneighborhood. Thus almost every trajectory returns arbitrarily close to where it started.

Suppose we have a collection of *N* classical atoms in a perfectly
sealed room. The phase-space dimension of this system is 6*N*. A
point in this phase space is denoted *z*. Suppose initially all the
atoms are, say, within one centimeter of one corner, with arbitrarily
chosen finite velocities. This corresponds to some initial point
*z*_{0} in the phase space. The phase space of the system is limited in
space by the room and in momentum by energy conservation; the phase
space is bounded. The recurrence theorem then says that in the
neighborhood of *z*_{0} there is an initial condition of the system that
returns to the neighborhood of *z*_{0} after some time. For the
individual atoms this means that after some time all of the atoms will
be found in the corner of the room again, and again, and again. Makes
one wonder about the second law of thermodynamics, doesn't it?^{32}

Some systems have attractors. An *attractor* is a region of phase
space that gobbles volumes of trajectories. For an attractor there
is some larger region, the basin of attraction, such that sets of
trajectories with nonzero volume eventually end up in the attractor
and never leave it. The recurrence theorem shows that Hamiltonian
systems with bounded phase space do not have attractors. Consider
some candidate volume in the proposed basin of attraction. The
recurrence theorem guarantees that some trajectories in the candidate
volume return to the volume repeatedly. Therefore, the volume is not
in a basin of attraction. Attractors do not exist in Hamiltonian
systems with bounded phase space.

This does not mean that every trajectory always returns. A simple example is the pendulum. Suppose we take a blob of trajectories that spans the separatrix, the trajectory that asymptotically approaches the unstable equilibrium with the pendulum pointed up. Trajectories with more energy than the separatrix make a full loop around and return to their initial point; trajectories with lower energy than the separatrix oscillate once across and back to their initial position; but the separatrix trajectory itself leaves the initial region permanently, and continually approaches the unstable point.

The definition of a dissipative system is not so clear. For some, ``dissipative'' implies that phase-space volume is not conserved, which is the same as saying the evolution of the system is not governed by Hamilton's equations. For others, ``dissipative'' implies that friction is present, representing loss of energy to unmodeled degrees of freedom. Here is a curious example. The damped harmonic oscillator is the paradigm of a dissipative system. Here we show that the damped harmonic oscillator can be described by Hamilton's equations and that phase-space volume is conserved.

The damped harmonic oscillator is governed by the ordinary differential equation

where is a coefficient of damping. We can formulate this
system with the Lagrangian^{33}

The Lagrange equation for this Lagrangian is

Since the exponential is never zero this equation has the same trajectories as equation (3.159) above.

The momentum conjugate to *x* is

For this system, the Hamiltonian is not the sum of the kinetic energy
of the motion of the mass and the potential energy stored in the
spring. The value of the Hamiltonian is not conserved
(_{0} *H* ne 0).
Hamilton's equations are

Let's consider a numerical case. Let *m* = 5, *k* = 1/4,
= 3. Here
the characteristic roots of the linear constant-coefficient ordinary
differential equation (3.159) are *s* = `-` 1/10, `-` 1/2. Thus the solutions are

for *A*_{1} and *A*_{2} determined by the initial conditions

Thus we can form the transformation from the initial state to the final state:

The transformation is linear, so the area is transformed by the determinant, which is 1 in this case. Thus, contrary to intuition, the phase-space volume is conserved. So why is this not a contradiction with the statement that there are no attractors in Hamiltonian systems? The answer is that the Poincaré recurrence argument is true only for bounded phase spaces. Here, the momentum expands exponentially with time (as the coordinate contracts), so it is unbounded.

We shouldn't really be too surprised by the way the theory protects itself from an apparent paradox -- that the phase volume is conserved even though all trajectories decay to zero velocity and coordinates. The proof of Liouville's theorem allows time-dependent Hamiltonians. In this case we are able to model the dissipation by just such a time-dependent Hamiltonian.

**Exercise 3.12.** **Time-dependent systems**

To make the fact that Liouville's theorem holds for time-dependent
systems even more concrete, extend the results of
section 3.8 to show how a swarm of initial points
outlining an area in the phase space of the *driven* pendulum
deforms as it evolves. Construct pictures analogous to
figures 3.25
and 3.26 for one of the interesting cases
where we have surfaces of section. Does the distortion look different
in different parts of the phase space? How?

We know the state of a system only approximately. It is reasonable to model our state of knowledge by a probability density function on the set of possible states. Given such incomplete knowledge, what are the probable consequences? As the system evolves, the density function also evolves. Liouville's theorem gives us a handle on this kind of problem.

Let *f*(*t*, *q*, *p*) be a probability density function on the phase space
at time *t*. For this to be a good probability density function we
require that the integral of *f* over all coordinates and momenta be
1 -- it is certain that the system is somewhere.

There is a set of trajectories that pass through any particular region
of phase space at a particular time. These trajectories are
neither created nor destroyed, and they proceed as a bundle to another
region of phase space at a later time. Liouville's theorem tells us
that the volume of the source region is the same as the volume of the
target region, so the density must remain constant. Thus
*D*(*f*o) = 0. If we have a system
described by the Hamiltonian *H* then

Since this must be true at each moment and since there is a solution
trajectory that emanates from every point in phase space, we may
abstract from solution paths and deduce a constraint on *f*:

This linear partial differential equation governs the evolution of the density function, and thus shows how our state of knowledge evolves.

^{31} The separatrix is the curve that separates the oscillating
motion from the circulating motion. It is made up of several
trajectories that are asymptotic to the unstable equilibrium.

^{32} It is reported that when Boltzmann was confronted with this
problem he responded, ``You should wait that long!''

^{33} This is just the product of the
Lagrangian for the undamped harmonic oscillator with an increasing
exponential of time.