Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 5: Lie Transformations Nonlinear Single-Particle Dynamics in High Energy Accelerators There are six lectures in this course on nonlinear dynamics: 1. First example: nonlinear dynamics in a bunch compressor 2. Second example: nonlinear dynamics in storage rings 3. Hamiltonian mechanics 4. Canonical perturbation theory 5. Lie transformations 6. Symplectic integrators Nonlinear Dynamics 1 Part 5: Lie Transformations In the previous lectures... We have seen how nonlinear dynamics can play an important role in some diverse and common accelerator systems. Nonlinear effects have to be taken into account when designing such systems. A number of powerful tools for analysis of nonlinear systems are developed from Hamiltonian mechanics. We have reviewed the principles of Hamiltonian mechanics in the context of accelerator beam dynamics, and have looked in particular at symplecticity, canonical transformations, and generating functions. Nonlinear Dynamics 2 Part 5: Lie Transformations In this lecture... In the final two lectures, we shall discuss methods for constructing transfer maps for accelerator elements. We have already seen how to do this for a drift space: but this is a special case, because the equations of motion can be solved exactly. In this lecture, we shall discuss two powerful techniques for constructing (and representing) maps for accelerator elements: • Lie transformations; • mixed-variable generating functions. We shall use a sextupole as an example, but the techniques we develop are quite general. Nonlinear Dynamics 3 Part 5: Lie Transformations Hamiltonian for a drift space Recall the general Hamiltonian for an accelerator element: v u !2 u 1 qφ 1 t 2 2 H = −(1 + hx) + δ − − (px − ax) − (py − ay) − 2 2 β0 P0c β0γ0 δ −(1 + hx)as + : (1) β0 For a drift space, this becomes: v u !2 u 1 1 δ t 2 2 H = − + δ − px − py − 2 2 + : (2) β0 β0γ0 β0 The equations of motion are given by Hamilton's equations: dx @H dpx @H = ; = − ; (3) ds @px ds @x and similarly for (y; py) and (z; δ). Nonlinear Dynamics 4 Part 5: Lie Transformations Transfer map for a drift space The equations of motion for a drift space are easy to solve, because the momenta px, py and δ are constants of the motion. The solution can be expressed as a map in closed form: the Hamiltonian is integrable. For the transverse variables: px0 x1 = x0 + ∆s; px1 = px0; (4) ps py0 y1 = y0 + ∆s; py1 = py0: (5) ps And for the longitudinal variables, we have: 0 1 + δ1 1 β0 z1 = z0 + @ − A ∆s; δ1 = δ0: (6) β0 ps The value of ps (a constant of the motion) is given by: v u !2 u 1 1 t 2 2 ps = + δ0 − px0 − py0 − 2 2: β0 β0γ0 Nonlinear Dynamics 5 Part 5: Lie Transformations Hamiltonian for a sextupole A sextupole field can be derived from the vector potential: 1 P0 3 2 Ax = 0;Ay = 0;As = − k2 x − 3xy : (7) 6 q This potential gives the fields: P0 1 P0 2 2 Bx = k2xy; By = k2 x − y ;Bs = 0: (8) q 2 q Note that the sextupole strength k2 is given by: q @2B = y (9) k2 2 : P0 @x The normalised potential ~a is given by: q ~a = A:~ (10) P0 Nonlinear Dynamics 6 Part 5: Lie Transformations Hamiltonian for a sextupole Hence, the Hamiltonian for a sextupole can be written: v u !2 u 1 1 1 δ t 2 2 3 2 H = − + δ − px − py − 2 2 + k2 x − 3xy + : β0 β0γ0 6 β0 (11) Since the coordinates x and y appear explicitly in the Hamiltonian, the momenta px and py are not constants. The equations of motion are nonlinear, and rather complicated. We will not even bother to write them down, since we cannot find an exact solution in closed form: the Hamiltonian is not integrable. To track a particle through a sextupole, we have to take one of two approaches: 1. integrate the equations of motion numerically (e.g. using a Runge{Kutta algorithm) with given initial conditions, or, 2. make some approximations that will enable us to write down an approximate map in closed form. Nonlinear Dynamics 7 Part 5: Lie Transformations Lie transformations Numerical techniques, such as Runge{Kutta algorithms, for integrating equations of motion are standard. The drawback in their use for accelerator beam dynamics is that they tend to be rather slow. Often, we are interested in tracking tens of thousands of particles, thousands of times around storage rings consisting of thousands of elements. Numerical integration is no good for this. We shall therefore focus on the second approach. We shall make some approximations that will enable us to write down a map in closed form. There are various ways to do this: we begin by developing the idea of Lie transformations. Lie transformations provide a means to construct a transfer map in closed form, even from a Hamiltonian that is not integrable. It is necessary to make some approximations, and these need to be understood in some detail. Nonlinear Dynamics 8 Part 5: Lie Transformations Lie operators Suppose we have a function f of the phase space variables, coordinates ~q and conjugate momenta ~p: f = f(~q; ~p): (12) Suppose we evaluate f at the location in phase space for a particle whose dynamics are governed by a Hamiltonian H. The time evolution of f is: df d~q @f d~p @f = + : (13) dt dt @~q dt @~p Using Hamilton's equations, this becomes: df @H @f @H @f = − : (14) dt @~p @~q @~q @~p Nonlinear Dynamics 9 Part 5: Lie Transformations Lie operators We define the Lie operator :g: for any function g(~q; ~p): @g @ @g @ :g: = − : (15) @~q @~p @~p @~q Constructing a Lie operator from the Hamiltonian, we can write: df = −:H: f: (16) dt Equation (16) suggests that given f at time t = t0, we can evaluate f at any later time t0 + ∆t: −∆t :H: f(t0 + ∆t) = e f : (17) t=t0 Here, the exponential of the Lie operator is defined in terms of a series expansion: ∆t2 ∆t3 e−∆t :H: = 1 − ∆t:H: + :H:2 − :H:3 + ··· (18) 2 3! Nonlinear Dynamics 10 Part 5: Lie Transformations Lie operators Equation (16) does indeed give us the value of f at any time t0 + ∆t, given the value of f at t = t0. We can see this by simply making a Taylor series expansion: df ∆t2 d2f ∆t3 d3f ( +∆ ) = ( )+ ∆ + + + f t0 t f t0 t 2 3 ··· dt t=t 2 dt 3! dt 0 t=t0 t=t0 (19) Then, since from (16) we can write: d = −:H: (20) dt equation (18) follows. Nonlinear Dynamics 11 Part 5: Lie Transformations Lie transformation The operator e:g: is called a Lie transformation. To see how to use Lie transformations to solve the equations of motion for a given system, consider a familiar example: a simple harmonic oscillator in one degree of freedom. The Hamiltonian is: 1 1 H = p2 + !2q2: (21) 2 2 Suppose we want to find the coordinate q as a function of time t. Of course, in this case, we could simply write down the equations of motion (from Hamilton's equations) and solve them (because the Hamiltonian is integrable). However, we can also write: −∆t :H: q(t0 + ∆t) = e q : (22) t=t0 Nonlinear Dynamics 12 Part 5: Lie Transformations Lie transformation example: harmonic oscillator To evaluate the Lie transformation, we need :H:q. This is given by (15): @H @q @H @q @H :H:q = − = − = −p: (23) @q @p @p @q @p Similarly, we find: :H:p = !2q: (24) This means that: :H:2q = :H:(−p) = −!2q; (25) :H:3q = :H:(−q) = !2p; (26) and so on. Nonlinear Dynamics 13 Part 5: Lie Transformations Lie transformation example: harmonic oscillator Using the above results, we find: ∆t2 q(t + ∆t) = qj − ∆t:H:qj + :H:2q − · · · 0 t=t0 t=t0 2 t=t0 (27) ∆t2 = q(t ) + ∆t p(t ) − !2 q(t ) − · · · (28) 0 0 2 0 Collecting together even and odd powers of t, we find that equation (28) can be written: p(t0) q(t0 + ∆t) = q(t0) cos(!∆t) + sin(!∆t): (29) ! Similarly (an exercise for the student!) we find that: −∆t :H: p(t0+∆t) = e p(t0) = −!q(t0) sin(!∆t)+p(t0) cos(!∆t): t=t0 (30) Nonlinear Dynamics 14 Part 5: Lie Transformations Lie transformation example: harmonic oscillator Equations (29) and (30) are the solutions we would have found using the conventional approach of integrating the equations of motion: but note that we have not performed any integrations, only differentiations (though we have had to sum an infinite series...) Lie operators and Lie transformations have many interesting properties that makes them useful for analysing the behaviour of dynamical systems. We shall explore these properties further in the next lecture; but for now, we shall simply see how to apply the technique demonstrated for the harmonic oscillator, to a particle moving through a sextupole. Nonlinear Dynamics 15 Part 5: Lie Transformations Sextupole map: Lie transformation approach Recall the Hamiltonian for a sextupole (11): v u !2 u 1 1 1 δ t 2 2 3 2 H = − + δ − px − py − 2 2 + k2 x − 3xy + : β0 β0γ0 6 β0 Using Lie operator notation, we can write the map for a particle moving through the sextupole as: −∆s :H: ~x(s0 + ∆s) = e ~x : (31) s=s0 Since the Lie transformation evolves the dynamical variables according to Hamilton's equations (for the Hamiltonian H) the map expressed in the form (31) is necessarily symplectic.