Structure-preserving techniques in accelerator physics

Abstract

To very good approximation, particularly for hadron machines, charged-particle trajectories in accelerators obey Hamiltonian mechanics. During routine storage times of eight hours or more, such particles execute some ${10}^8$ revolutions about the machine, ${10}^{10}$ oscillations about the design orbit, and ${10}^{13}$ passages through various bending and focusing elements. Prior to building, or modifying, such a machine, we seek to identify accurately the long-term behaviour and stability of particle orbits over such large numbers of interactions. This demanding computational effort does not yield easily to traditional methods of symplectic numerical integration, including both explicit Yoshida-type and implicit Runge–Kutta or Gaussian methods. As an alternative, one may compute an approximate one-turn map and then iterate that map. We describe some of the essential considerations and techniques for constructing such maps to high order and for realistic magnetic field models. Particular attention is given to preserving the symplectic condition characteristic of Hamiltonian mechanics.

Keywords:

• Symplectic map
• symplectic jet
• geometric integration
• lie algebra
• Poincaré generating function
• cremona map

2020 AMS SUBJECT CLASSIFICATIONS:

• 37M15
• 37J11
• 65P10
• 70H15

Acknowledgments

We are grateful to the U.S. Department of Energy Office of Science for research support over the years on the use of Map and Lie-Algebraic methods in Accelerator Physics. In addition, we thank RadiaSoft LLC for partial support provided to one of us (D.T.A.) during the preparation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 But, even with the highest achievable beam densities, beam-beam collisions are sufficiently rare that the stored beams are only partially depleted over the storage time. However, the colliding beams can have significant dynamical (both linear and nonlinear) effects on each other, a complication that must be understood/managed but lies beyond the scope of this chapter.

2 Much of the background material for this chapter is most easily found on the Web in a draft book: Dragt [5]. It, in turn, provides numerous additional references. In subsequent citations it will be referred to as LM. For a discussion of Lagrangians and Hamiltonians for charged-particle motion in electromagnetic fields, see LM, Sections 1.5–1.7.

3 We are embarrassed by the custom of also using the symbol q to denote the charge of the particle in question.

4 LM, Section 1.6.

5 LM, Section 1.6 and Exercise 1.6.7

6 The quantity $(x^{\mathrm{\prime }}{)}^{\mu }{A}^{\nu }{g}_{\mu \nu }$ is a scalar under Lorentz transformations provided the 4-potential $A^{\nu }$ actually transforms as a 4-vector. See [LM, footnotes in Exercise 1.6.7.] for a discussion of the contrary case.

7 Again see [LM, footnotes in Exercise 1.6.7.] for a discussion of the contrary case.

8 When using $H_R$, we employ only solutions that obey (25(25) $x^{\mathrm{\prime }}\cdot x^{\mathrm{\prime }}=1.$(25) ).

9 Here, by the ‘time’, we mean whatever has been selected to be the independent variable. Note also that the symbol z now no longer refers to the third component of $\mathit{r}$, but rather to the collection of phase-space variables.

10 LM, Section 1.3.

11 In what follows, the letters sp and Sp are used as abbreviations for symplectic.

12 LM, Subsection 6.4.1.

13 LM, Section 12.9.

14 LM, Section 5.1.

15 LM, Section 6.1.2.

16 LM, Section 8.2.

17 LM, Section 7.6.

18 The letters m and M that appear in this and the previous sentence are abbreviations for map. The letters i and I are abbreviations for inhomogeneous. By inhomogeneous it is meant that the possibility of constant terms appearing in (48(48) $z_a^f=\sum _b{R}_{ab}{z}_b^i+\sum _{bc}{T}_{abc}{z}_b^i{z}_c^i+\sum _{bcd}{U}_{abcd}{z}_b^i{z}_c^i{z}_d^i+\cdots ,$(48) ) is included.

19 Note that a similar optimism is shared by practitioners of symplectic integration.

20 LM, Chapters 17-25.

21 LM, Subsections 1.2.3, 1.4.1, 1.4.2, and Exercise 1.4.3.

22 It follows from (36(36) $M^{T}JM=J,$(36) ) that a symplectic map must preserve Poisson brackets, and vice versa.

23 LM, Section 10.5 and Chapter 39. Some authors refer to TPSA as Differential Algebra (DA). For an exposition of DA, see Berz [3].

24 Symplectic completion of symplectic jets using a generating function was first implemented—in the context of Accelerator Physics—in the Lie-algebra based accelerator design code MaryLie.

25 LM, Section 6.7.

26 LM, Chapter 34. See also the article Erdélyi and Berz [7]. Currently we do not find persuasive their invocation of the Hofer metric, but do agree (for other reasons) with their conclusion that use of the Poincaré generating function has several desirable features.

27 Surprisingly, the terms through degree 3 in the Taylor expansions of (76) agree with the terms generated by applying (63(63) ${{\mathcal{N}}^{}}^{\mathrm{tr}3}={\mathcal{I}}^{}+\text{:}q{p}^2\text{:}+\frac{1}{2}{\text{:}q{p}^2\text{:}}^2.$(63) ) to $(q,p)$. This happens due to the second relation in (80(80) $F_{+}^3(u)=-\frac{1}{4}{f}_3(u)\mathrm{and}{F}_{+}^4(u)=-\frac{1}{8}{f}_4(u).$(80) ) and our tacit assumption that $f_4=0$.

28 LM, Section 34.4.

29 The iteration process does converge more rapidly for q, p values that lie closer to the origin, but the improvement is not impressive. For example, after reducing the distance from the origin by a factor of 5, at least 12 iterations are still required to achieve convergence to machine precision.

30 Detailed background material for this section is most easily found on the Web: Abell [1]. In subsequent citations it will be referred to as TMCA. See also the chapter Dragt and Abell [6]. See further Blanes [4]. For a history of the term Cremona maps, and for a fuller discussion of the concepts of kicks and jolts that we present here (including the extension to two and three degrees of freedom), see TMCA, Chapters 10 and 11.

31 LM, section 7.3.

32 The ${\sigma }_j^r$ do depend on l, but we have suppressed this index to avoid notational clutter.

33 In the case of two or three degrees of freedom, this becomes $\sum _j{w}_j{\sigma }_j^r{\sigma }_j^s$, with weights $w_j$ differing from 1/N.

34 TMCA, Section 16.1.1

35 TMCA, Section 16.1.3

36 TMCA, Sections 16.1.4 and 16.1.6

37 LM, Section 3.7.3

38 LM, Section 34.3.5

39 LM, Sections 4.1 and 10.8

40 Cubature formulas are higher-dimensional analogs of quadrature formulas.

41 See TMCA, Part II. Also see Abell, McIntosh, and Schmidt [2].