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Orbits in Particle Accelerators and the Completion of Symplectic Jets

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Orbits in Particle Accelerators and the Completion of Symplectic Jets Orbits in particle accelerators and the completion of symplectic jets Ivo Bodin 10372164 8 july 2015 1 Contents 1 Summary 3 2 Introduction 3 3 An introduction to symplectomorphisms 4 3.1 Group of linear symplectomorphisms . 4 3.2 Group of symplectomorphisms . 6 3.3 Introduction to kick maps . 8 4 Introduction to canonical transformations 11 4.1 Notation . 11 4.2 Lagrange equations of motion . 12 4.3 Hamilton's equations of motion . 15 4.4 Relation canonical transformation and symplectomorphisms . 16 4.5 Liouville's theorem . 17 4.6 Generating functions . 19 5 Completion of symplectic jets using kick maps 21 6 Completion of symplectic jets using generating functions 28 7 Comparison 33 2 1 Summary CERN wants to know how to complete a symplectic jet. This thesis describes two different ways to do this. One is a purely mathematical approach. This approach delivers an exact way to solve the problem. However, it turns out that it is not possible to perform this method using the computers we have today. The other approach uses theory of classical mechanics. A numerical approximation is required. This approach turns out to be useful in practice. 2 Introduction In 1954 twelve countries in Western Europe started CERN. Today most people know CERN for their particle accelerator, the Large Hadron Collider, mostly known as LHC. A particle accelerator uses electromagnetic fields to accelerate particles to high speeds. The LHC is so-called synchrotron. This means it is cyclic. That is all a mathematician needs to know about particle accelerators to read this thesis. In this thesis I assume that the particles follow Hamiltonian mechanics. This means we threat the system in a non-relativistic way. CERN would like to know the path that particles take. To do this, they measure the particles' place and momentum at a given point in the accelerator. This yields six coordinates. Two place coordinates, three momenta coordinates and the time. In this thesis, we look at three place coordinates and three momenta coordinates. If the length of the particle accelerator is known, we can calculate the forward momentum from the time and vice versa. These six coordinates are measured every time the particle completes a lap. CERN would like to predict what the next coordinates would be given a set of coordinates. This is an unknown function of six variables. Hamiltonian mechanics dictate that this function is a symplectomorphism. CERN can measure this function up to a certain precision. In practice they can measure all terms of the power series up to order eleven. This measured function will, in general, not be a symplectomorphism. We will call this function a symplectic jet. Experiments show that a symplectic jet will diverge if the function is iterated. Experiments also show that symplectomorphisms tend to diverge a lot slower than a similar symplectic jets. This is not a topic of this thesis. We will take it as a given. Because of this, CERN would like to find a symplectomorphism that agrees with the measured function up to order eleven. We call this process the completion of a symplectic jet. A.J. Dragt and D.T. Abell have written an article describing this phenomenon and suggesting ways to complete a symplectic jet. In this thesis, two of their most promising ways are worked out in detail. First is a pure mathematical approach. This gives an exact completion of a symplectic jet. However, the symplectomorphism it produces is too big to work with in practice. The second way uses 3 generating functions. This way need numerical approximation but is doable in practice. Both approaches are both proven to be correct and a few advantages and disadvantages are named. 3 An introduction to symplectomorphisms 3.1 Group of linear symplectomorphisms In this thesis, we look at function from R2n to R2n. Symplectomorphisms are not defined for odd dimensional spaces. We will first give the definition of linear symplectomorphisms. 2n 2n Definition 1. Let Ψ be a linear mapping from R to R and define J0 by 2 3 0 1 6 7 J0 = 6 7 4 5 −1 0 Then Ψ is a linear symplectomorphism if and only if T Ψ J0Ψ = J0 Lemma 1. The set of linear symplectomorphisms S from R2n to R2n form a group. T T Proof. Let Ψ; Φ 2 S. Then, by definition:Ψ J0Ψ = J0 and Φ J0Φ = J0. We calculate T T T (ΨΦ) J0(ΨΦ) =Φ (Ψ J0Ψ)Φ T =Φ J0Φ =J0 We conclude that, by definition, ΨΦ 2 S. Therefore we can conclude that S is closed. We will now show that the identity matrix In 2 S. By definition T In J0In = J0 2n 2n This proves that In 2 S. Since In is the identity for linear function from R to R , it also serves as the identity for the set of linear symplectomorphisms. We will now prove that Ψ 2 S implies that Ψ−1 2 S. Let Ψ 2 S. Then, by definition: T Ψ J0Ψ = J0. Doing a simple calculation −1 T −1 −1 T T −1 (Ψ ) J0Ψ =(Ψ ) Ψ J0ΨΨ T −1 T −1 =(Ψ ) Ψ J0ΨΨ =J0 4 Therefore, we can conclude that Ψ−1 2 S. Since taking compositions of functions is an associative operation, the set of linear symplectomorphisms is associative given this operation. We conclude that the set of all linear symplectomorphisms from R2n to R2n satisfies all axioms of a group. Definition 2. We define the group of linear symplectomorphisms from R2n to R2n T SymL(2n; R) = fM 2 GL2n(R)jM J0M = J0g From this definition, it is evident that SymL(2n; R) ⊂ GL2n(R) The group of linear symplectomorphisms has another less evident property, namely SymL(2n; R) ⊂ SL(2n; R) In other words, an element M of the group of linear symplectomorphisms has the property det(M) = 1 This can be proven using a polynomial called the Pfaffian of the matrix. However, in the canonical transformation section of this thesis, I will prove that the determinant of the Ja- cobian of every canonical transformation equals one. This implies that all linear symplecto- morphisms have a determinant that equals one. It is not much work to show that the absolute value of the determinant of a linear symplec- tomorphism equals one. Using known properties of the determinant yields T det(J0) =det(Ψ J0Ψ) T =det(Ψ )det(J0)det(Ψ) 2 =det(Ψ) det(J0) We can conclude that det(Ψ)2 = 1. Direct from the definition of a linear symplectomorphism, we can find a set of equations that will be useful to find other properties of SymL(2n; R). We write a linear map in a similar way as the matrix J0. We calculate. 2 3T 2 3 2 3 2 3 2 3 AB 0 1 AB AT CT CD 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 = 6 7 6 7 4 5 4 5 4 5 4 5 4 5 CD −1 0 CD BT DT −A −B 2 3 AT C − CT AAT D − BCT 6 7 = 6 7 4 5 BT C − ADT BT D − DT B 5 We conclude that a linear map is a linear symplectomorphism if and only if: AT C − CT A =0 BT D − DT B =0 T T A D − BC =In This set of equation helps us to calculate the dimension of SymL(2n; R). From the first equation, we can deduce AT C =CT A =(AT C)T This implies that AT C is symmetrical. This demand constraints half of the off-diagonal T 1 elements of A C, for a total of 2 n(n − 1). The second demand constraints an equal amount of elements. The last demand constraints n2 elements. We conclude that the dimension of SymL(2n; R) is equal to 1 dim(GL (R)) − 2 · n(n − 1) − n2 = (2n)2 − n(n − 1) − n2 2n 2 = 2n2 + n For n = 1, the first two restrictions are empty. The third restriction reduces to det(Ψ) = 1. Therefore, SymL(2; R) =SL(2; R). This is not true for n > 1. This can easily be seen in the following calculation dim(SL(2n; R)) =3n2 dim(SymL(2; R)) =2n2 + n For n 6= 1, the dimensions are different. Therefore, the groups cannot be equal. It will however stay true that SymL(2; R) ⊂ SL(2n; R) 3.2 Group of symplectomorphisms We will now extend the definition to the definition of a linear symplectomorphism to the definition of a symplectomorphism Definition 3. Let f be a total differentiable map from R2n to R2n. Then f is a symplecto- morphism if and only if the Jacobian matrix Df satisfies T (Df) J0Df = J0 6 If f is a linear symplectomorphism, then Df is a symplectic matrix. Therefore this definition is equivalent to the first definition for linear mappings. We will now define a symplectic jet. Definition 4. Let f be a total differentiable map from R2n to R2n. Then f is a symplectic jet of order m if and only if the Jacobian matrix Df satisfies T m+1 (Df) J0Df − J0 = O(z ) We will now prove an equivalent statement to check if something is a symplectomorphism. Lemma 2. Let f be a total differentiable map from R2n to R2n. Let !(u; v) be defined as T !(u; v) = u Jov Then f is a symplectomorphism if and only if for all u; v 2 R2n !(u; v) = Df ∗(!(u; v)) := !(Df(u); Df(v)) We call f ∗! the pullback of f.
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