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Symplectic Tracking RoutineMalte Titze, Helmholtz-Zentrum Berlin, 10.05.2014
1. Introduction
2. Main Idea
3. Advantages
4. Theory
5. Summary
0. Overview
Z
X
Y
Given: (Ax, A
y, A
z, φ) as a Fourier-decomposition with respect to the
(longitudinal) Z-axis.
1. Introduction
Z
Y
Given: (Ax, A
y, A
z, φ) as a Fourier-decomposition with respect to the
(longitudinal) Z-axis.
X
1. Introduction
Z
X
Y
z0
zf
How to effectively track particles symplectic from z0 to z
f?
1. Introduction
Z
X
Y
z0
zf
Find the dependency of the cartesian coordinates to the cyclic ones at the final position z
f (time-independent case).
2. Main Idea
Z
X
Y
z0
zf
Find the dependency of the cartesian coordinates to the cyclic ones at the final position z
f (time-independent case).
Functions of (x0, y
0, p
x0, p
y0), z
0 and z
f
2. Main Idea
Based on the following data:
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. The final position zf.
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. The final position zf.
→ (xf, y
f, p
xf, p
yf) can be computed to arbitrary precision without the need
of a PDE-solver!
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. The final position zf.
→ (xf, y
f, p
xf, p
yf) can be computed to arbitrary precision without the need
of a PDE-solver!
→ The coordinate transformation is symplectic.
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. The final position zf.
→ (xf, y
f, p
xf, p
yf) can be computed to arbitrary precision without the need
of a PDE-solver!
→ The coordinate transformation is symplectic.
→ Fringe fields are included.
3. Advantages
Based on the following data:
1. The initial coordinates (x0, y
0, p
x0, p
y0) and z
0,
2. The values of the field at the initial coordinates (Fourier-coefficients),
3. The final position zf.
→ (xf, y
f, p
xf, p
yf) can be computed to arbitrary precision without the need
of a PDE-solver!
→ The coordinate transformation is symplectic.
→ Fringe fields are included.
→ There are analytic formulas of the fields in the case of multipoles.
3. Advantages
The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A
x, A
y, A
z, φ) can be written as
4. Theory
The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A
x, A
y, A
z, φ) can be written as
4. Theory
The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A
x, A
y, A
z, φ) can be written as
with
4. Theory
The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A
x, A
y, A
z, φ) can be written as
with
Note that in this description, the Z-component will play the role as the 'time' and (t, -E) is a new pair of conjugate variables.
4. Theory
This means, the equations of motion have the form:
4. Theory
This means, the equations of motion have the form:
4. Theory
This means, the equations of motion have the form:
It follows especially:
for the kicks in X- and Y-direction.
4. Theory
In the following we assume that
1. all fields are time-independent.
4. Theory
In the following we assume that
1. all fields are time-independent.
2. no electric fields.
4. Theory
In the following we assume that
1. all fields are time-independent.
2. no electric fields.
3. the kicks are small enough, so that products of order two and higher can be neglected.
4. Theory
In the following we assume that
1. all fields are time-independent.
2. no electric fields.
3. the kicks are small enough, so that products of order two and higher can be neglected.
Assumption 3 is not necessary in order to make the method work, it merely simplifies the Hamiltonian. Higher orders can be included.
4. Theory
In the following we assume that
1. all fields are time-independent.
2. no electric fields.
3. the kicks are small enough, so that products of order two and higher can be neglected.
Assumption 3 is not necessary in order to make the method work, it merely simplifies the Hamiltonian. Higher orders can be included.
Dropping assumptions 1 and/or 2 will have a deeper impact on the theory.
4. Theory
Exclude in the radicand of the Hamiltonian
4. Theory
Exclude in the radicand of the Hamiltonian
and develop the square root, using the small angular approximation:
4. Theory
Exclude in the radicand of the Hamiltonian
and develop the square root, using the small angular approximation:
where we introduced the normalized quantities
4. Theory
Note: By introducing this new Hamiltonian , the equations of motions for the X- and Y-coordinates will not change in these approximations:
4. Theory
Note: By introducing this new Hamiltonian , the equations of motions for the X- and Y-coordinates will not change in these approximations:
4. Theory
Note: By introducing this new Hamiltonian , the equations of motions for the X- and Y-coordinates will not change in these approximations:
In the following we will drop all tilde symbols again.
4. Theory
A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v
x, v
y, z) satisfying
4. Theory
A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v
x, v
y, z) satisfying
4. Theory
A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v
x, v
y, z) satisfying
and
4. Theory
A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v
x, v
y, z) satisfying
and
inserting px and p
y into this last equation gives the partial differential
equation for F we are going to solve.
4. Theory
Hence, the partial differential equation for F has the form:
4. Theory
Hence, the partial differential equation for F has the form:
where we redefined the magnetic potentials by a parameter epsilon to provide a measure of the field strength.
4. Theory
Hence, the partial differential equation for F has the form:
where we redefined the magnetic potentials by a parameter epsilon to provide a measure of the field strength.
In the absence of any fields, this differential equation can be solved directly:
in which is a constant.
4. Theory
This leads to
4. Theory
This leads to
and from the last equation we get
Similar equations hold for the Y-component. This corresponds to a free drift.
4. Theory
This leads to
and from the last equation we get
Similar equations hold for the Y-component. This corresponds to a free drift.
The canonical momenta px and p
y can be converted to the kinetic
momenta using the vector potentials and the normalization factor introduced earlier.
4. Theory
In the general case, we make the following perturbative ansatz:
4. Theory
In the general case, we make the following perturbative ansatz:
where fijk are functions of x, y and z.
4. Theory
In the general case, we make the following perturbative ansatz:
where fijk are functions of x, y and z.
We enter the general differential equation with this ansatz and compare coefficients, using its special nature:
4. Theory
This yields the following system of equations for the fijk's:
4. Theory
This yields the following system of equations for the fijk's:
with
The semicolon indicates a partial derivative with respect to the corresponding coordinate.
4. Theory
This yields the following system of equations for the fijk's:
with
The semicolon indicates a partial derivative with respect to the corresponding coordinate.
Note that the left-hand side of the above equation is determined by functions of lower total order i + j + k and the potentials up to a function h
ijk
of x and y, the 'integration constant'.
4. Theory
The equations of the smallest total orders have the form
4. Theory
Let us see the implication of this ansatz for the momenta:
4. Theory
Let us see the implication of this ansatz for the momenta:
4. Theory
Let us see the implication of this ansatz for the momenta:
and similar for the Y-component:
4. Theory
Let us see the implication of this ansatz for the momenta:
and similar for the Y-component:
This means: If we fix the functions hijk(x, y) by the condition f
ijk(x, y, z
f) ≡ 0,
then we get
and we can invert the above system of equations (for px and p
y) at ε = 1
by a Newton-iteration to get pxf and p
yf.
4. Theory
Once we have computed the values pxf and p
yf, we can determine the
offset xf by
4. Theory
Once we have computed the values pxf and p
yf, we can determine the
offset xf by
4. Theory
Once we have computed the values pxf and p
yf, we can determine the
offset xf by
and similarly for the Y-component. Again we have set ε = 1.
4. Theory
Once we have computed the values pxf and p
yf, we can determine the
offset xf by
and similarly for the Y-component. Again we have set ε = 1.
The kicks at the final position are computed by
4. Theory
We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks.
5. Summary
We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks.
Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order.
5. Summary
We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks.
Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order.
The time-dependent case and the inclusion of electric fields will however alter the differential equation. It is an open question of how to implement a perturbative generating function in these cases.
5. Summary
We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks.
Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order.
The time-dependent case and the inclusion of electric fields will however alter the differential equation. It is an open question of how to implement a perturbative generating function in these cases.
Another interesting subject are the generalization of the method to density distributions.
Thank you for your attention!
5. Summary