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