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The most important scalar function for a dynamical system is its
dynamical phase. This phase is the (value of) the integral,
Eq.(1), evaluated for a worldline
which starts at
, which satisfies the Lagrange equations of motion,
and which therefore extremizes this integral. Let us designate this
extremal value by
Figure 3:
Five different world lines
having the same starting point and terminating at points where the
dynamical phase along the world lines has reached the value .
The locus of such termination points of such world lines with common
starting point forms the isogram of a scalar function
, the dynamical phase (``HamiltonJacobi function'')
of the Hamiltonian system.

This integral is a function of the worldline's termination point which
we take to be
. If there is only one worldlines between
and
, as is usually the case, the
is a single valued function of the termination point.
This is depicted in Figure 3. (If there
were several such worldlines, then would be multivalued.) Thus is a
function of the location
, ,
of that termination point. It also is a function of the
parameter which one uses to parametrize the worldline. Thus
one has
Next: The HamiltonJacobi Equation for
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Ulrich Gerlach
20051107