6.4 Conserved quantities, kinetic energy density
The fact that Equations (11) – (12) are wave-map equations has some important consequences. In
particular, since the target has a three-dimensional isometry group, there should be three conserved
quantities. In fact, there are constants , and such that
That the derivatives of the right-hand sides of these expressions are zero can be verified using
Equations (11) – (12).
Another important consequence of the wave-map structure is the fact that isometries of hyperbolic space
map solutions to solutions.
Finally, it is convenient to introduce the kinetic energy density,
Note that this quantity is obtained by computing the time derivative of the wave map and then taking the
squared length of the resulting vector field using the target metric. As a consequence, the kinetic energy is a
geometric object. In other words, two solutions related by an isometry have the same kinetic energy