In this section we demonstrate that time evolution generates a canonical transformation: if we consider all possible initial states of a Hamiltonian system and follow all the trajectories for the same time interval, then the map from the initial state to the final state of each trajectory is a canonical transformation.

We use time evolution to generate a transformation

that is obtained in the following way. Let (*t*) = (*t*, (*t*),
(*t*)) be a solution of Hamilton's equations. The transformation
_{} satisfies

Notice that _{} changes the time component. This is the
first transformation of this kind that we have considered.^{24}

Given a state (*t*', *q*', *p*'), we find the phase-space path
emanating from this state as an initial condition, satisfying

The value (*t*, *q*, *p*) of _{}(*t*', *q*', *p*') is then
(*t*' + , (*t*' + ), (*t*' + )).

Time evolution is canonical if the transformation _{}
is symplectic and if the Hamiltonian transforms in an appropriate
manner. The transformation _{} is symplectic if the
bilinear antisymmetric form is invariant (see
equation 5.73) for a general pair of
linearized state variations with zero time component.

Let ' be an increment with zero time component of the state
(*t*', *q*', *p*'). The linearized increment in the value of _{}(*t*', *q*', *p*') is = *D*_{}(*t*', *q*', *p*')
'. The image of the increment is obtained by multiplying the
increment by the derivative of the transformation. On the other hand,
the transformation is obtained by time evolution, so the image of the
increment can also be found by the time evolution of the linearized
variational system.
Let

be variations of the state path (*t*) = (*t*, (*t*), (*t*));
then

This must be true for arbitrary , so it is satisfied if the following quantity is constant:

With Hamilton's equations, the variations satisfy

Substituting these into *D**A* and collecting terms, we find^{25}

We conclude that time evolution generates a phase-space transformation with symplectic derivative.

To make a canonical transformation we must specify how the Hamiltonian transforms. The same Hamiltonian describes the evolution of a state and a time-advanced state because the latter is just another state. Thus the transformed Hamiltonian is the same as the original Hamiltonian.

We deduced that volumes in phase space are preserved by time evolution by showing that the divergence of the phase flow is zero, using the equations of motion (see section 3.8). We can also show that volumes in phase space are preserved by the evolution using the fact that time evolution is a canonical transformation.

We have shown that phase-space volume is preserved for symplectic transformations. Now we have shown that the transformation generated by time evolution is a symplectic transformation. Therefore, the transformation generated by time evolution preserves phase-space volume. This is an alternate proof of Liouville's theorem.

There is another canonical transformation that can be constructed from
time evolution. We define the transformation '_{} such
that

where *S*_{}(*a*, *b*, *c*) = (*a* + , *b*, *c*) shifts the time
of a phase-space state.^{26}
More explicitly, given a state (*t*, *q*', *p*'), we evolve the state
that is obtained by subtracting from *t*; that is, we take the state
(*t* `-` , *q*', *p*') as an initial state for evolution by Hamilton's
equations. The state path satisfies

The output of the transformation is the state

The arguments of '_{} are not a consistent phase-space
state; the time argument must be decremented by to obtain a
consistent state. The transformation is completed by evolution of
this consistent state.

Why is this a good idea? Our usual canonical transformations do not change the time component. This modified time-evolution transformation is thus of the form discussed previously. The resulting time-evolution transformation is canonical and in the usual form:

This transformation can also be extended to
be a canonical transformation,
with an appropriate adjustment of the Hamiltonian. The Hamiltonian
*H*'_{} that gives the correct Hamilton's equations at the
transformed phase-space point is the original Hamiltonian composed
with a function that decrements the independent variable by :

Notice that if *H* is time independent then *H*'_{} = *H*.

Assume we have a procedure `C` such that `((C delta-t) state)`
implements a time-evolution transformation _{} of the state `state`
with time interval `delta-t`;
then the procedure `Cp` such that `((Cp delta-t) state)` implements
'_{} of the same state and time interval can be derived from the procedure `C` by using the procedure

`(define ((C->Cp C) delta-t)
(compose (C delta-t) (shift-t (- delta-t))))
`

where `shift-t` implements *S*_{}:

`(define ((shift-t delta-t) state)
(up
(+ (time state) delta-t)
(coordinate state)
(momentum state)))
`

To complete the canonical transformation we have a procedure that transforms the Hamiltonian:

`(define ((H->Hp delta-t) H)
(compose H (shift-t (- delta-t))))
`

So both and ' can be used to make canonical
transformations by specifying how the old and new Hamiltonians are
related. For _{} the Hamiltonian is unchanged. For
'_{} the Hamiltonian is time shifted.

**Exercise 5.21.** **Verification**

The condition (5.19) that Hamilton's
equations are preserved for _{} is

and the condition that Hamilton's
equations are preserved for '_{} is

Verify that these conditions are satisfied.

**Exercise 5.22.** **Driven harmonic oscillator**

We can use the simple driven harmonic oscillator to illustrate that
time evolution yields a symplectic transformation that can be
extended to be canonical in two ways. We use the driven harmonic
oscillator because its solution can be compactly expressed in explicit
form.

Suppose that we have a harmonic oscillator with natural frequency
_{0} driven by a periodic sinusoidal drive of frequency
and amplitude . The Hamiltonian we will consider is

The general solution for a given initial state (*t*_{0}, *q*_{0}, *p*_{0})
evolved for a time is

where ' = /(_{0}^{2} `-` ^{2}).

**a**. Fill in the details of the procedure

`(define (((C alpha omega omega0) delta-t) state)
... )
`

that implements the time-evolution transformation of the driven harmonic oscillator.

**b**. In terms of `C`, the general solution emanating
from a given state is

`(define (((solution alpha omega omega0) state0) t)
(((C alpha omega omega0) (- t (time state0))) state0))
`

Check that the implementation of `C` is correct by using it
to construct the solution and verifying that the solution satisfies
Hamilton's equations. Further check the solution by comparing to
numerical integration.

**c**. We know that for any phase-space state function *F* the rate of change
of that function along a solution path is

Show, by writing a short program to test it, that this is true of the
function implemented by `(C delta)` for the driven oscillator.
Why is this interesting?

**d**. Verify that both `C` and `Cp` are symplectic using
`symplectic?`.

**e**. Use the procedure `canonical?` to verify that
both `C` and `Cp` are canonical with the appropriate
transformed Hamiltonian.

We can also show that time evolution generates canonical transformations using the Poincaré-Cartan integral invariant.

Consider a two-dimensional region of phase-space coordinates, *R*', at
some particular time *t*' (see figure 5.8). Let *R* be the
image of this region at time *t* under time evolution for a time
interval of . The time evolution is governed by a Hamiltonian
*H*. Let sum_{i} *A*_{i} be the sum of the oriented areas of the
projections of *R* onto the fundamental canonical planes.^{27}
Similarly, let sum_{i} *A*'_{i} be the sum of oriented projected areas for
*R*'. We will show that sum_{i} *A*_{i} = sum_{i} *A*'_{i}, and thus the
Poincaré integral invariant is preserved by time evolution. By
showing that the Poincaré integral invariant is preserved, we will
have shown that the *q**p* part of the transformation generated by time
evolution is symplectic. From this we can construct canonical
transformations from time evolution as before.

In the extended phase space we see that the evolution sweeps out a
cylindrical volume with endcaps *R*' and *R*, each at a
fixed time. Let *R*'' be the two-dimensional region swept out by the
trajectories that map the boundary of region *R*' to the boundary of
region *R*. The regions *R*, *R*', and *R*'' together form the
boundary of a volume of phase-state space.

The Poincaré-Cartan integral invariant on the whole boundary is
zero.^{28}
Thus

where the *n* index indicates the *t*, *p*_{t} canonical plane. The second
term is negative, because in the extended phase space we take the area to
be positive if the normal to the surface is outward pointing.

We will show that the Poincaré-Cartan integral invariant for a region of phase space that is generated by time evolution is zero:

This will allow us to conclude

The areas of the projection of *R* and *R*' on the *t*, *p*_{t} plane are
zero because *R* and *R*' are at constant times, so for these regions
the Poincaré-Cartan integral invariant is the same as the Poincaré
integral invariant. Thus

We are left with showing that the Poincaré-Cartan integral invariant
for the region *R*'' is zero. This will be zero if the contribution
from any small piece of *R*'' is zero. We will show this by showing
that the form (see equation 5.71) on a small
parallelogram in this region
is zero. Let (0; *q*, *t*; *p*, *p*_{t}) be a vertex of this parallelogram. The
parallelogram is specified by two edges _{1} and _{2}
emanating from this vertex with components (0; *q*, *t*;
*p*, *p*_{t}). For edge _{1} of the parallelogram, we take
a constant-time phase-space increment with length *q* and
*p* in the *q* and *p* directions. The first-order change in
the Hamiltonian that corresponds to these changes is

for constant time *t* = 0. The increment *p*_{t} is the
negative of *H*. So the extended phase-space increment is

The edge _{2} is obtained by time evolution of the vertex for a
time interval *t*. Using Hamilton's equations, we obtain

The form applied to these incremental states that form the edges of this parallelogram gives the area of the parallelogram:

So we may conclude that the integral of this expression over the entire surface of the tube of trajectories is also zero. Thus the Poincaré-Cartan integral invariant is zero for any region that is generated by time evolution.

Having proven that the trajectory tube provides no contribution, we have
shown that the Poincaré integral invariant of the two endcaps is the
same. This proves that time evolution generates a symplectic *q**p*
transformation.

We can use the Poincaré-Cartan invariant to prove that for autonomous two-degree-of-freedom systems surfaces of section (constructed appropriately) preserve area.

To show this we consider a surface of section for one coordinate (say
*q*_{2}) equal to zero. We construct the section by accumulating the
(*q*_{1}, *p*_{1}) pairs. We assume that all initial conditions have
the same energy. We compute the sum of the areas of canonical
projections in the extended phase space again. Because all initial
conditions have the same *q*_{2} = 0 there is no area on the *q*_{2},
*p*_{2} plane, and because all the trajectories have the same value of the
Hamiltonian the area of the projection on the *t*, *p*_{t} plane is also
zero. So the sum of areas of the projections is just the area of the
region on the surface of section. Now let each point on the surface
of section evolve to the next section crossing. For each point on the
section this may take a different amount of time. Compute the sum of
the areas again for the mapped region. Again, all points of the
mapped region have the same *q*_{2} so the area on the *q*_{2}, *p*_{2}
plane is zero, and they continue to have the same energy so the area on
the *t*, *p*_{t} plane is zero. So the area of the mapped region is again
just the area on the surface of section, the *q*_{1}, *p*_{1} plane. Time
evolution preserves the sum of areas, so the area on the surface of
section is the same as the mapped area.

So surfaces of section preserve area provided that the section points
are entirely on a canonical plane. For example, to make the
Hénon-Heiles surfaces of section (see
section 3.6.3)
we plotted *p*_{y} versus *y* when *x* = 0 with *p*_{x} __>__ 0. So
for all section points the *x* coordinate has the fixed value 0, the
trajectories all have the same energy,
and the points accumulated are entirely in the *y*, *p*_{y} canonical
plane. So the Hénon-Heiles surfaces of section preserve area.

We can show that time evolution generates a symplectic transformation directly from the action principle.

Recall that the Lagrangian action *S* is

We computed the variation of the action in deriving the Lagrange equations. The variation is (see equation 1.33)

rewritten in terms of the Euler-Lagrange operator *E*. In the
derivation of the Lagrange equations we considered only variations
that preserved the endpoints of the path being tested. However,
equation (5.347) is true of arbitrary variations.
Here we consider variations that are not zero at the endpoints
around a realizable path *q* (one for which
*E* [ *L* ] o [*q*] = 0 ). For these variations
the variation of the action is just the integrated term:

Recall that *p* and are structures, and the product implies a
sum of products of components.

Consider a continuous family of realizable paths; the path for parameter *s* is
(*s*) and the coordinates of this path at time *t* are
(*s*)(*t*).
We define (*s*) = *D*(*s*); the variation of the path
along the family is the
derivative of the parametric path with respect to the parameter.
Let

be the value of the action from *t*_{1} to *t*_{2} for path (*s*).
The derivative of the action along this parametric family of paths is^{29}

Because (*s*) is
a realizable path, *E*[*L*] o [(*s*)] = 0. So

where (*s*) is the conjugate momentum to (*s*).
The integral of *D* is

In conventional notation the latter line integral is written

where _{1}(*s*) = (*s*)(*t*_{1}) and _{2}(*s*) = (*s*)(*t*_{2}).

For a loop family of paths (such that (*s*_{2}) = (*s*_{1})), the
difference of actions at the endpoints vanishes, so we deduce

which is the line-integral version of the integral invariants.

In terms of area integrals, using Stokes's theorem, this is

where *R*^{i}_{j} are the regions in the *i*th canonical plane.
We have found that the time evolution preserves the integral
invariants, and thus time evolution generates a symplectic transformation.

^{24} Our theorems about which transformations are canonical are
still valid, because they required only that the derivative of the
independent variable be 1.

^{25} Partial derivatives of structured arguments do not generally
commute, so this deduction is not as simple as it may appear. It is
helpful to introduce component indices and consider the equation
componentwise.

^{26} The transformation *S*_{} is an identity on the *q**p*
components, so it is symplectic. Although it adjusts the time, it
is not a time-dependent transformation in that the *q**p* components
do not depend upon the time. Thus, if we adjust the Hamiltonian by
composition with *S*_{} we have a canonical transformation.

^{27} By Stokes's theorem we may compute the area of a region by a line
integral around the boundary of the region. We define the
positive sense of the area to be the area enclosed
by a curve that is traversed in a counterclockwise direction,
when drawn on a plane with the coordinate on the abscissa and the
momentum on the ordinate.

^{28} We can see this as follows.
Let be any closed curve in the boundary. This curve divides
the boundary into two regions. By Stokes's theorem the integral
invariant over both of these pieces can be written as a line integral
along this boundary, but they have opposite signs, because is
traversed in opposite directions to keep the surface on the left. So
we conclude that the integral invariant over the entire surface is zero.

^{29} Let *f* be a path-dependent function,
(*s*) = *D*(*s*), and *g*(*s*) =
*f*[(*s*)]. The variation of *f* at (*s*)
in the direction (*s*) is
_{(s)} *f*[(*s*)] = *D* *g*(*s*).