Let us imagine that a spin-orbit transformation
such that:
type normal_spin type(normalform) N ! regular orbital normal form type(damapspin) a1 ! brings to fixed point type(damapspin) ar ! normalises around the fixed point type(damapspin) as ! pure spin map type(damapspin) a_t ! !! (a_t%m,a_t%s) = (a1%m, I ) o (I ,as%s) o (ar%m,I) !!! extra spin info integer M(NDIM,NRESO),MS(NRESO),NRES ! orbital and spin resonances to be left in the map type(real_8) n0(3) ! n0 vector type(real_8) theta0 ! angle for the matrix around the orbit (analogous to linear tunes) !!!Envelope radiation stuff real(dp) s_ij0(6,6) ! equilibrium beam sizes ! equilibrium emittances (partially well defined only for infinitesimal damping) real(dp) emittance(3),tune(3),damping(3) logical(lp) AUTO,STOCHASTIC real(dp) KICK(3) ! fake kicks for tracking stochastically real(dp) STOCH(6,6) ! Diagonalized of stochastic part of map for tracking stochastically end type normal_spin
If NS is of type normal_spin and DS is a damapspin, then DS can be normalized as follows:
NS=DSin complete analogy with the orbital normal form. In fact the analogy on the computer is complete. 11.1
So, as I said, if you believe in the normal form of Eq. (11.10), then even a stupid person can comprehend the concept of the invariant spin axis in the normalized space.
In the normalized space, we notice that the spin vector is an invariant of . Of course the choice of for the spin normal form is arbitrary. In the rarified air of mathematics, we are free to use anything for the normal form axis. We chose a ``vertical'' axis. Further more the angle of the spin rotation is itself invariant because it depends only on the actions and the parameters of the orbital map . So the spin axis is defined.
We now intend to show the obvious: the vector is the spin field axis .