68
W. Herr and E. Forest
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
px
x
Exact quadrupole versus thin lens approximation
Exact map and non-symplectic map
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
px
x
Exact quadrupole versus thin lens approximation
Exact map and symplectic map O(2)
Fig. 3.3 Poincare section for tracking through a quadrupole. Comparison between exact solution,
non-symplectic (left) and symplectic (right) tracking. Shown are symplectic integrators of order 1
and 2
use of high order integrators. Secondly, models which require self-consistent time
tracking or s tracking (e.g. space charge calculations) must use integrators for which
s(t) is monotonic in the magnets.
3.6.3.4 Comparison Symplectic Versus Non-symplectic Integration
A demonstration of a non-symplectic tracking is shown in Fig. 3.3. A particle is
tracked through a quadrupole and the poincare section is shown. A quadrupole
is chosen because it allows a comparison with the exact solution. The nonsymplecticity causes the particle to spiral outwards. As comparison to the exact
tracking is shown. In Fig. 3.3 (right) the symplectic integrators of order 1 and
2 as derived above are used instead. The trajectory is now constant and the
difference to the exact solution is small. Although the model is approximated
but symplectic, the underlying physics (i.e. constant energy in this case)
is correct at the expense of a small discrepancy with respect to the exact
solution.
3.7 Hamiltonian Treatment of Electro-Magnetic Fields
A frequently asked question is why one should not just use Newton’s laws and the
Lorentz force. Some of the main reasons are:
– Newton requires rectangular coordinates and time, trajectories with e.g. “curvature” or “torsion” need to introduce “reaction forces”. (For example: LHC has
locally non-planar (cork-screw) “design” orbits!).
– For linear dynamics done by ad hoc introduction of new coordinate frame.
W. Herr and E. Forest
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
px
x
Exact quadrupole versus thin lens approximation
Exact map and non-symplectic map
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
px
x
Exact quadrupole versus thin lens approximation
Exact map and symplectic map O(2)
Fig. 3.3 Poincare section for tracking through a quadrupole. Comparison between exact solution,
non-symplectic (left) and symplectic (right) tracking. Shown are symplectic integrators of order 1
and 2
use of high order integrators. Secondly, models which require self-consistent time
tracking or s tracking (e.g. space charge calculations) must use integrators for which
s(t) is monotonic in the magnets.
3.6.3.4 Comparison Symplectic Versus Non-symplectic Integration
A demonstration of a non-symplectic tracking is shown in Fig. 3.3. A particle is
tracked through a quadrupole and the poincare section is shown. A quadrupole
is chosen because it allows a comparison with the exact solution. The nonsymplecticity causes the particle to spiral outwards. As comparison to the exact
tracking is shown. In Fig. 3.3 (right) the symplectic integrators of order 1 and
2 as derived above are used instead. The trajectory is now constant and the
difference to the exact solution is small. Although the model is approximated
but symplectic, the underlying physics (i.e. constant energy in this case)
is correct at the expense of a small discrepancy with respect to the exact
solution.
3.7 Hamiltonian Treatment of Electro-Magnetic Fields
A frequently asked question is why one should not just use Newton’s laws and the
Lorentz force. Some of the main reasons are:
– Newton requires rectangular coordinates and time, trajectories with e.g. “curvature” or “torsion” need to introduce “reaction forces”. (For example: LHC has
locally non-planar (cork-screw) “design” orbits!).
– For linear dynamics done by ad hoc introduction of new coordinate frame.