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As an example, consider the problem of solving the H-J equation for
a charged particle in the vector potential
|
(12) |
It expresses a generic plane wave traveling towards positive . The
H-J equation is
To make this equation physically more transparent and mathematically
more manageable one introduces the retarded and advanced time
coordinates
and
In terms of these the H-J equation becomes
|
(13) |
This equation is readily solved by the method of separation of
variables according to which the solution,
is a sum of antiderivatives, each one depending only on
its respective integration variable. With this stipulation one finds
that the components of the gradient of are
where , , , as well as , are the constants of
separation1. Consequently, the
Hamilton-Jacobi function (``Hamilton's principal function'',
``Schroedinger phase'', dynamical phase)
|
(19) |
has the form
or more generally
Both and are solutions to the H-J equation, but has an
additive constant
which is a function
of the four separation constants.
Although the difference between and is trivial from the
perspective of solving the H-J equation, the opposite is true from the
viewpoint of physics and mathematics. Indeed, suppose one let's
Then one obtains
The usefulness of this H-J function is that it generates new
phase space coordinates relative to which the solutions of Hamilton's
Eqs.(3)-(4) are straight lines. They are depicted in
Figure 8 on page . In other words, solving the H-J equation for
is tantamount to solving Hamilton's equations for all possible initial
conditions.
Next: Dynamical Phase as Physical
Up: Laser-driven particle mechanics
Previous: The Hamilton-Jacobi Equation for
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Ulrich Gerlach
2005-11-07