3 Non-linear Dynamics in Accelerators
63
The physical meaning of this condition is that the map is area preserving in the
phase space. The condition can easily be derived from a Hamiltonian treatment,
closely related to Liouville’s theorem.
3.6.2 Approximations and Tools
The concept of symplecticity is vital for the treatment of Hamiltonian systems. This
is true in particular when the stability of a system is investigated using particle
tracking. However, in practice it is difficult to accomplish for a given exact problem.
As an example we may have the exact fields and potentials of electromagnetic
elements. For a single pass system a (slightly) non-symplectic integrator may be
sufficient, but for an iterative system the results are meaningless.
To track particles using the exact model may result in a non-symplectic tracking,
i.e. the underlying model is correct, but the resulting physics is wrong.
It is much better to approximate the model to the extend that the tracking is
symplectic. One might compromise on the exactness of the final result, but the
correct physics is ensured.
As a typical example one might observe possible chaotic motion during the
tracking procedure. However, there is always a non-negligible probability that this
interpretation of the results may be wrong. To conclude that it is not a consequence
of non-symplecticity of the procedure or a numerical artifact it is necessary to
identify the physical mechanism leading to this observation.
This may not be possible to achieve using the exact model as input to a
(possibly) non-symplectic procedure. Involving approximations to the definition of
the problem should reveal the correct physics at the expense of a (hopefully) small
error. Staying exact, the physics may be wrong.
As a result, care must be taken to positively identify the underlying process.
This procedure should be based on a approximations as close as possible to the
exact problem, but allowing a symplectic evaluation.
An example for this will be shown in Sect. 3.6.3.4.
3.6.3 Taylor and Power Maps
A non-linear element cannot be represented in the form of a linear matrix and more
complicated maps have to be introduced [5]. In principle, any well behaved, nonlinear function can be developed as a Taylor series. This expansion can be truncated
at the desired precision.
Another option is the representation as Lie transformations [8, 10]. Both types
are discussed in this section.
63
The physical meaning of this condition is that the map is area preserving in the
phase space. The condition can easily be derived from a Hamiltonian treatment,
closely related to Liouville’s theorem.
3.6.2 Approximations and Tools
The concept of symplecticity is vital for the treatment of Hamiltonian systems. This
is true in particular when the stability of a system is investigated using particle
tracking. However, in practice it is difficult to accomplish for a given exact problem.
As an example we may have the exact fields and potentials of electromagnetic
elements. For a single pass system a (slightly) non-symplectic integrator may be
sufficient, but for an iterative system the results are meaningless.
To track particles using the exact model may result in a non-symplectic tracking,
i.e. the underlying model is correct, but the resulting physics is wrong.
It is much better to approximate the model to the extend that the tracking is
symplectic. One might compromise on the exactness of the final result, but the
correct physics is ensured.
As a typical example one might observe possible chaotic motion during the
tracking procedure. However, there is always a non-negligible probability that this
interpretation of the results may be wrong. To conclude that it is not a consequence
of non-symplecticity of the procedure or a numerical artifact it is necessary to
identify the physical mechanism leading to this observation.
This may not be possible to achieve using the exact model as input to a
(possibly) non-symplectic procedure. Involving approximations to the definition of
the problem should reveal the correct physics at the expense of a (hopefully) small
error. Staying exact, the physics may be wrong.
As a result, care must be taken to positively identify the underlying process.
This procedure should be based on a approximations as close as possible to the
exact problem, but allowing a symplectic evaluation.
An example for this will be shown in Sect. 3.6.3.4.
3.6.3 Taylor and Power Maps
A non-linear element cannot be represented in the form of a linear matrix and more
complicated maps have to be introduced [5]. In principle, any well behaved, nonlinear function can be developed as a Taylor series. This expansion can be truncated
at the desired precision.
Another option is the representation as Lie transformations [8, 10]. Both types
are discussed in this section.