Suppose we have a system with *n* + 1 degrees of freedom described by a
time-independent Hamiltonian in a (2*n* + 2)-dimensional phase space.
Here we can play the converse game: we can choose any generalized
coordinate to play the role of ``time'' and the negation of its
conjugate momentum to play the role of a new *n*-degree-of-freedom
time-dependent Hamiltonian in a *reduced phase space* of 2*n*
dimensions.

and suppose we have a system described by a time-independent Hamiltonian

For each solution path there is a conserved quantity *E*.
Let's choose a coordinate *q*^{n} to be the time in a reduced phase
space. We define
the dynamical variables for the *n*-degree-of-freedom reduced phase space:

In the original phase space a coordinate such as *q*^{n} maps time to a
coordinate. In the formulation of the reduced phase space we will
have to use the inverse function = (*q*^{n})^{-1} to map the coordinate to
the time, giving the new coordinates in terms of the new time

We propose that a Hamiltonian in the reduced phase
space is the negative of the inverse of *f*(*q*^{0}, ..., *q*^{n}; *p*_{0}, ...,
*p*_{n}) = *E* with respect to the *p*_{n} argument:

Note that in the reduced phase space we will have indices for the
structured variables in the range 0 `...` *n* `-` 1, whereas in the original
phase space the indices are in the range 0 `...` *n*. We will show
that *H*_{r} is an appropriate Hamiltonian for the given dynamical
system in the reduced phase space. To compute Hamilton's equations we
must expand the implicit definition of *H*_{r}. We define an auxiliary
function

Note that *by construction* this function is identically a
constant *g* = *E*. Thus all of its partial derivatives are zero:

where we have suppressed the arguments.
Solving for partials of *H*_{r}, we get

Using these relations, we can deduce the Hamilton's equations in the reduced phase space from the Hamilton's equations in the original phase space:

Consider planar motion in a central field. We have already seen this expressed in polar coordinates in equation (3.99):

There are two degrees of freedom and the Hamiltonian is
time independent. Thus the energy, the value of the Hamiltonian, is
conserved on realizable paths.
Let's forget about time and
reparameterize this system in terms of the orbital radius *r*.^{16}
To do this we solve

which is the Hamiltonian in the reduced phase space.

Hamilton's equations are now quite simple:

We see that *p*_{} is independent of *r* (as it was with *t*), so for
any particular orbit we may define a constant angular momentum *L*.
Thus our problem ends up as a simple quadrature:

To see the utility of this procedure, we continue our example with a definite potential energy -- a gravitating mass point:

When we substitute this into equation (5.139) we obtain a mess that can be simplified to

Integrating this, we obtain another mess, which can be simplified and rearranged to obtain the following:

This can be recognized as the polar-coordinate form of the equation of
a conic section with eccentricity *e* and parameter *p*:

In fact, if the orbit is an ellipse with semimajor axis *a*, we have

and so we can identify the role of energy and angular momentum in shaping the ellipse:

What we get from analysis in the reduced phase space is the geometry of the trajectory, but we lose the time-domain behavior. The reduction is often worth the price.

Although we have treated time in a special way so far, we have found that time is not special. It can be included in the coordinates to make a driven system autonomous. And it can be eliminated from any autonomous system in favor of any other coordinate. This leads to numerous strategies for simplifying problems, by removing time variation and then performing canonical transforms on the resulting conservative autonomous system to make a nice coordinate that we can then dump back into the role of time.

^{16} We
could have chosen to reparameterize in terms of , but then both
*p*_{r} and *r* would occur in the resulting time-independent
Hamiltonian. The path we have chosen takes advantage of the fact that
does not appear in our Hamiltonian, so *p*_{} is a constant
of the motion. This structure suggests that to solve this kind of
problem we need to look ahead, as in playing chess.