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W. Herr and E. Forest
3.1.1 Motivation
The most reliable tools to study (i.e. description of the machine) are simulations
(e.g. tracking codes).
• Particle Tracking is a numerical solution of the (nonlinear) Initial Value Problem.
It is a “integrator” of the equation of motion and a vast amount of tracking codes
are available, together with analysis tools (Examples: Lyapunov, Chirikov, chaos
detection, frequency analysis, . . . )
• It is unfortunate that theoretical and computational tools exist side by side
without an undertaking how they can be integrated.
• It should be undertaken to find an approach to link simulations with theoretical
analysis, would allow a better understanding of the physics in realistic machines.
• A particularly promising approach is based on finite maps [1].
3.1.2 Single Particle Dynamics
The concepts developed here are used to describe single particle transverse dynamics in rings, i.e. circular accelerators or storage rings. This is not a restriction for the
application of the presented tools and methods. In the case of linear betatron motion
the theory is rather complete and the standard treatment [2] suffices to describe the
dynamics. In parallel with this theory the well known concepts such as closed orbit
and Twiss parameters are introduced and emerged automatically from the CourantSnyder formalism [2]. The formalism and applications are found in many textbooks
(e.g. [3–5]).
In many new accelerators or storage rings (e.g. LHC) the description of the
machine with a linear formalism becomes insufficient and the linear theory must be
extended to treat non-linear effects. The stability and confinement of the particles is
not given a priori and should rather emerge from the analysis. Non-linear effects are
a main source of performance limitations in such machines. A reliable treatment is
required and the progress in recent years allows to evaluate the consequences. Very
useful overview and details can be found in [6–8].
3.1.3 Layout of the Treatment
Following a summary of the sources of non-linearities in circular machine, the basic
methods to evaluate the consequences of non-linear behaviour are discussed. Since
the traditional approach has caused misconception and the simplifications led to
wrong conclusions, more recent and contemporary tools are introduced to treat
these problems. An attempt is made to provide the physical picture behind these
tools rather than a rigorous mathematical description and we shall show how the
W. Herr and E. Forest
3.1.1 Motivation
The most reliable tools to study (i.e. description of the machine) are simulations
(e.g. tracking codes).
• Particle Tracking is a numerical solution of the (nonlinear) Initial Value Problem.
It is a “integrator” of the equation of motion and a vast amount of tracking codes
are available, together with analysis tools (Examples: Lyapunov, Chirikov, chaos
detection, frequency analysis, . . . )
• It is unfortunate that theoretical and computational tools exist side by side
without an undertaking how they can be integrated.
• It should be undertaken to find an approach to link simulations with theoretical
analysis, would allow a better understanding of the physics in realistic machines.
• A particularly promising approach is based on finite maps [1].
3.1.2 Single Particle Dynamics
The concepts developed here are used to describe single particle transverse dynamics in rings, i.e. circular accelerators or storage rings. This is not a restriction for the
application of the presented tools and methods. In the case of linear betatron motion
the theory is rather complete and the standard treatment [2] suffices to describe the
dynamics. In parallel with this theory the well known concepts such as closed orbit
and Twiss parameters are introduced and emerged automatically from the CourantSnyder formalism [2]. The formalism and applications are found in many textbooks
(e.g. [3–5]).
In many new accelerators or storage rings (e.g. LHC) the description of the
machine with a linear formalism becomes insufficient and the linear theory must be
extended to treat non-linear effects. The stability and confinement of the particles is
not given a priori and should rather emerge from the analysis. Non-linear effects are
a main source of performance limitations in such machines. A reliable treatment is
required and the progress in recent years allows to evaluate the consequences. Very
useful overview and details can be found in [6–8].
3.1.3 Layout of the Treatment
Following a summary of the sources of non-linearities in circular machine, the basic
methods to evaluate the consequences of non-linear behaviour are discussed. Since
the traditional approach has caused misconception and the simplifications led to
wrong conclusions, more recent and contemporary tools are introduced to treat
these problems. An attempt is made to provide the physical picture behind these
tools rather than a rigorous mathematical description and we shall show how the