Longitudinal position latexmath:[z] of a particle relatice to the reference particle [m].
Longitudinal position latexmath:[z] of a particle relative to the reference particle [m].
DELTA::
latexmath:[\frac{E}{P_0 c}-\frac{1}{\beta_0}] Energy derviation [1].
latexmath:[\frac{E}{P_0 c}-\frac{1}{\beta_0}] Energy derivation [1].
(Where latexmath:[E] is the total energy of the particle and latexmath:[\beta_0 = \frac{u}{c}] the speed relative to the speed of light latexmath:[c] of the reference particle)
The independent variable is position of the reference particle *s* [m].
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@@ -71,7 +71,7 @@ latexmath:[\mathbf{\mathcal{M}}] is the map. This map can represent either a bea
[[sec.opalmap.creationOfMap]]
==== Creation of map
The creation of the element map is based on the Hamiltonian Mechanic, more specificly on the Lie Operator.
The creation of the element map is based on the Hamiltonian Mechanic, more specifically on the Lie Operator.
[latexmath]
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@@ -82,11 +82,11 @@ The creation of the element map is based on the Hamiltonian Mechanic, more speci
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Here latexmath:[H], the time dependent Hamiltonian represents the total energy, consisting of the kinetic latexmath:[T] and potential latexmath:[V] energy.
The lower equations are the Hamiltonian equations of motion, where the momenta latexmath:[p_i] and the positons latexmath:[q_i] form the cannonical paris.
Using cannonical transformations, the Hamiltonian can be adjusted to use the pathlength latexmath:[s] as independent and the particle parameters as depenedent variable(s).
The lower equations are the Hamiltonian equations of motion, where the momenta latexmath:[p_i] and the positions latexmath:[q_i] form the canonical pairs.
Using canonical transformations, the Hamiltonian can be adjusted to use the pathlength latexmath:[s] as independent and the particle parameters as dependent variable(s).
Introducing the Lie operator, which acts similar to a "waiting" Poissant Bracket:
Introducing the Lie operator, which acts similar to a "waiting" Poisson Bracket:
[latexmath]
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\begin{aligned}
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@@ -94,7 +94,7 @@ Introducing the Lie operator, which acts similar to a "waiting" Poissant Bracket
\end{aligned}
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If a function latexmath:[f:= f_{\left( \mathbf{q_{(s)}},\mathbf{p_{(s)}}\right) }] describes one of the phase space variables latexmath:[v] its total derivative to the independent variable, combined with the Hamiltonian equations of motions, is similar to the Lie operator times the independ variable, i. e. latexmath:[s].
If a function latexmath:[f:= f_{\left( \mathbf{q_{(s)}},\mathbf{p_{(s)}}\right) }] describes one of the phase space variables latexmath:[v] its total derivative to the independent variable, combined with the Hamiltonian equations of motions, is similar to the Lie operator times the indepedent variable, i. e. latexmath:[s].
[latexmath]
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@@ -119,10 +119,10 @@ Where latexmath:[e^{-:\!H\!: s}] is the Lie expansion.
[[sec.opalmap.Implementation]]
==== Implementaion of map tracking
==== Implementation of map tracking
For the derivative of the Hamiltonian, a Differential Algebra (DA) was used, in particular the Truncated Power Series Algebra (TPSA).
This algebra uses the Taylor expansion as the equivalent function, which also is responsible for its name by creating truncated power series.
Just form the definition of the Taylor expansion, it can be seen that a finite, to the order latexmath:[\Omega], expansion is an approximaion of the acutal function, due to the error term latexmath:[\mathcal{O}\left( \mathbf{v}_{\left( \Delta s\right)}^{\,\Omega +1}\right) ].
Just form the definition of the Taylor expansion, it can be seen that a finite, to the order latexmath:[\Omega], expansion is an approximation of the actual function, due to the error term latexmath:[\mathcal{O}\left( \mathbf{v}_{\left( \Delta s\right)}^{\,\Omega +1}\right) ].
[latexmath]
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@@ -131,7 +131,7 @@ Just form the definition of the Taylor expansion, it can be seen that a finite,
\end{aligned}
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In _OPAL-map_ the Hamiltonian gets Taylor expanded and its map derived (`link:opalmap#sec.opalmap.creationOfMap[Creation of Map]`) in the TPSA using the _OPAL_ DA package. The truncation length gets definded in `TRACK` setting the `MAP_ORDER` attribute.
In _OPAL-map_ the Hamiltonian gets Taylor expanded and its map derived (`link:opalmap#sec.opalmap.creationOfMap[Creation of Map]`) in the TPSA using the _OPAL_ DA package. The truncation length gets defined in `TRACK` setting the `MAP_ORDER` attribute.
