# Tracking in SAD

### Strategic Accelerator Design

- 6D full-symplectic tracking
- Dynamic aperture survey
- Synchrotron radiation

### Synchrotron radiation

SAD can track particles with synchrotron radiation. In this case, each magnet is split into several slices. Inside the slice, a particle obeys symplectic map and at each border of slices it loses momenta/energy randomly depending on the magnetic field felt by the particle. The random number is made according to the classical

radiation spectrum[4,5].

### 6D full-symplectic tracking

All maps in SAD is symplectic, except for radiation related maps. Even with the presence of radiation, the map from the Hamiltonian nature should be symplectic, because nobody has established a theorem on how much the map should be symplectic in the presence of radiation. Those who can stand with non-symplectic maps are not welcome to SAD (and to KEK).

### Dynamic aperture survey[3]

### key words:

**Symplecticity **

A (nonlinear) map $x\rightarrow f(x)$ is called symplectic when the Jacobi matrix

$M_{ij}\equiv\partial f_i/\partial x_j$ satisfies $MJM^t=J$. Here, $x_i$ stands for six

canonical variables $(x,p_x,y,p_y,z,p_z)$ and

J={{0,1,0,0,0,0},

{-1,0,0,0,0,0},

{0,0,0,1,0,0},

{0,0,-1,0,0,0},

{0,0,0,0,0,1},

{0,0,0,0,-1,0}}.

**Frequently, one might misunderstand that M is symplectic if det(M)=1. This is true only for dynamics in one degree of freedom.** But, it is true that det M=1 when M is symplectic[1].

A sufficient condition for M be symplectic is that there are finite number of functions

$g_1(x),g_2(x),\cdots,g_n(x)$ and $f(x)$ can de writtens in a form

$\exp:g_1: \exp:g_2: \cdots \exp:g_n:$

where $:g:$ is the Lie transformation[2].

### Lie Transformation[2]

Let F(x) be any function of the canonical variables x. The Lie transformation is defined by

\[ \exp:F:x=x+[F,x]+1/2[F,[F,x]]+\cdots, \]

where [f,g] is the Poisson bracket.

This is symplectic automatically. But if you trancate the expansion at a finite order, it looses

the symplcticity.

### Dynamic aperture[3]

It is one of the most fundamental and important objects in beam dynamics. However, nobody has succeeded to derivce the dynamic aperture from a given Hamiltonian. Even, it seems that there is no good constructive definition of the dynamic aperture.

### References

- 生出勝宣、ビーム力学入門、JLC-FFIR92研究会講義録・研究報告集、田内利明、山本昇編集、KEK Proceedings 93-6 (1993).
- A. J. Dragt, in Physics of High Energy Accelerators, proceedings of the Summer School on High Energy Particle Accelerators, Fermilab, 1981, edited by R.A. Carrigan, I.R. Huson, and M. Month (AIP Conf. Proc. No. 87) (AIP, New York, 1982), p. 147.
- K.Oide and H.Koiso,
*Dynamic aperture of electron storage rings with noninterleaved sextupoles,*Phys.Rev.**E47,**2010 (1993). - K.Oide,
*Synchrotron-Radiation Limit on the Focusing of Electron Beams,*Phys.Rev. Lett.**61,**1713 (1988). - K.Ohmi, K.Hirata and K.Oide,
*From the beam-envelope matrix to synchrotron-radiation integrals,*Phys.Rev.**E49**, 751 (1994).