6.4 Conserved quantities, kinetic energy density

The fact that Equations (11View Equation) – (12View Equation) 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 A, B and C such that
∫ 2P A = 1{2Q (tQt )e − 2(tPt )}d 𝜃 (25 ) ∫ S B = e2P(tQt)d𝜃 (26 ) ∫ S1 2P 2 C = {(tQt )(1 − e Q ) + 2Q (tPt)}d𝜃. (27 ) S1
That the derivatives of the right-hand sides of these expressions are zero can be verified using Equations (11View Equation) – (12View Equation).

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,

𝒦 = P 2τ + e2PQ2τ. (28 )
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 density.

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