Using the above theorem, it is easy to see that if a system P (D) is well posed, then so is the system P (D) + B, where B is any constant matrix. For the particular case at hand, this means that we can further restrict attention, without loss of generality, to the the principal part of the operator, namely
where
is a
matrix valued vector in
. In this case we can improve on the above condition by showing
that well posedness implies no growth of the solution, that is
that we can choose
above.
If
satisfies the above condition for some
, then we say that
P
(D) is
strongly hyperbolic, which, as we see, is equivalent for first order equation
systems to well posedness. If the operator
H
does not depend on
, a case that appears in most physical problems, then we say the
system is
symmetric hyperbolic
. Indeed, if
H
does not depend on
, then there is a base in which it just becomes the identity
matrix. (One can diagonalize it and re-scale the base.) Then the
above condition in the new base just means that
-with the upper matrix index lowered- is symmetric for any
, and so each component of
is symmetric. Even in the general (strongly hyperbolic) case,
one can find a base (
dependent) in which
can be diagonalized, basically because it is symmetric with
respect to the (
dependent) scalar product induced by
. In this diagonal version, it is easy to see that the well
posedness requires all eigenvalues of
to be purely imaginary. Thus we see that an equivalent
characterization for well posedness of first order systems is
that their principal part (i.e.
) has purely imaginary eigenvalues, and that it can be
diagonalized by an invertible,
-dependent, transformation. The classical example of a symmetric
hyperbolic system is the wave equation.
For simplicity we consider the wave equation in 1+1 dimensions. Choosing Cartesian coordinates we have,
and so defining the ``vector''
we have the following first order system:
There are several other notions of hyperbolicity that appear in the literature:
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |