Basics of beam dynamics 5
x
y
s
o
X
Y
Z
O
r 0 (s)
Figure 1.1 The curvilinear coordinate system for the description of beam motion.
orbit and the deviations from the reference orbit to measure the positions
and directions of the particles. This coordinate system is called Frenet-Serret
coordinate system, which is illustrated in Figure 1.1.
The design path is represented by a curve, r 0 (s), in the global Cartesian
coordinate system (O-XY Z), where s is the path length. At any point along
the design path, the position of a particle is measured in the local coordinate
system, o-xys, with the unit vectors along x, y, and s directions given as
follows: ˆ s is the tangent unit vector of the design path, ˆ
x is the normal unit
vector, and ˆ
y is the cross product of ˆ s and ˆ
x. x and y are the two transverse
directions and s is the longitudinal direction. Usually the design path is a
planar curve on the horizontal plane, hence x is referred to as the horizontal
direction and y the vertical direction. The design path could consist of sections
that bend in the vertical direction or in a plane that is at an arbitrary angle
with the horizontal plane. In such cases, the local coordinate system can rotate
about the s direction at the transition points in and out of those sections.
In general, the motion of charged particles in external electromagnetic
fields is governed by the Hamiltonian [68]
H = qΦ + c
m 2 c 2 + (P − qA) 2 ,
(1.1)
where c is the speed of light in vacuum, q and m are the charge and mass of the
particle, respectively, Φ and A are the scalar and vector potentials of the fields,
respectively, P = p + qA is the canonical momentum, and p is the mechanical
momentum. The transverse motion is typically determined by static magnetic
fields, for which Φ = 0 and
∂A
∂t = 0. In principle, after the vector potential is
specified over the space, the motion of a particle is completely determined by
the initial conditions of the position and the canonical momentum.
x
y
s
o
X
Y
Z
O
r 0 (s)
Figure 1.1 The curvilinear coordinate system for the description of beam motion.
orbit and the deviations from the reference orbit to measure the positions
and directions of the particles. This coordinate system is called Frenet-Serret
coordinate system, which is illustrated in Figure 1.1.
The design path is represented by a curve, r 0 (s), in the global Cartesian
coordinate system (O-XY Z), where s is the path length. At any point along
the design path, the position of a particle is measured in the local coordinate
system, o-xys, with the unit vectors along x, y, and s directions given as
follows: ˆ s is the tangent unit vector of the design path, ˆ
x is the normal unit
vector, and ˆ
y is the cross product of ˆ s and ˆ
x. x and y are the two transverse
directions and s is the longitudinal direction. Usually the design path is a
planar curve on the horizontal plane, hence x is referred to as the horizontal
direction and y the vertical direction. The design path could consist of sections
that bend in the vertical direction or in a plane that is at an arbitrary angle
with the horizontal plane. In such cases, the local coordinate system can rotate
about the s direction at the transition points in and out of those sections.
In general, the motion of charged particles in external electromagnetic
fields is governed by the Hamiltonian [68]
H = qΦ + c
m 2 c 2 + (P − qA) 2 ,
(1.1)
where c is the speed of light in vacuum, q and m are the charge and mass of the
particle, respectively, Φ and A are the scalar and vector potentials of the fields,
respectively, P = p + qA is the canonical momentum, and p is the mechanical
momentum. The transverse motion is typically determined by static magnetic
fields, for which Φ = 0 and
∂A
∂t = 0. In principle, after the vector potential is
specified over the space, the motion of a particle is completely determined by
the initial conditions of the position and the canonical momentum.