Rewrite (hyper.1) as a first order system

or equivalently,

The essential ingredient here is the existence of a positive symmetrizer, *H* > 0
,

so that multiplication by *H* on the left gives

Multiplying by we are led to

and the real part of both sides are in fact perfect derivatives, for by the symmetry of *H*,

and similarly, by the symmetry of , we have

Hence, by integration over the -period we end up with energy
conservation, asserting

We note that the positivity of *H* was not used in the proof and is assumed
just for
the sake of making (*u*,*Hu*) an admissible convex ``energy norm.''

Thu Jan 22 19:07:34 PST 1998