Synchro-Betatron Motion in Circular Accelerators

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1 Outlines Synchro-Betatron Motion in Circular Accelerators Kevin Li March 30, 2011 Kevin Li Synchro-Betatron Motion 1/ 70

2 Outline of Part I Outlines Part I: and Model Introduction Part II: The Transverse Hamiltonian Part III: The Synchro-Betatron Hamiltonian 1 Collective effects in the longitudinal plane Numerical and computational tools in accelerator physics 2 Basic physics Specialisation to classical electromagnetic theory 3 The symplectic structure Kevin Li Synchro-Betatron Motion 2/ 70

3 Outline of Part II Outlines Part I: and Model Introduction Part II: The Transverse Hamiltonian Part III: The Synchro-Betatron Hamiltonian 4 The Transverse Hamiltonian Canonical transformations Coordinate and rescaling transformation Transverse dynamics Kevin Li Synchro-Betatron Motion 3/ 70

4 Outline of Part III Outlines Part I: and Model Introduction Part II: The Transverse Hamiltonian Part III: The Synchro-Betatron Hamiltonian 5 The Synchro-Betatron Hamiltonian Series of canonical transformations RF fields The full Synchro-Betatron Hamiltonian Kevin Li Synchro-Betatron Motion 4/ 70

5 Part I and Model Introduction Kevin Li Synchro-Betatron Motion 5/ 70

6 Outline Collective effects in the longitudinal plane Numerical and computational tools 1 Collective effects in the longitudinal plane Numerical and computational tools in accelerator physics 2 Basic physics Specialisation to classical electromagnetic theory 3 The symplectic structure Kevin Li Synchro-Betatron Motion 6/ 70

7 SPS ecloud effects Collective effects in the longitudinal plane Numerical and computational tools Frozen synchrotron motion: Dynamics: 17.6e10 protons, 1e12 electrons Tune footprint: Kevin Li Synchro-Betatron Motion 7/ 70

8 SPS ecloud effects Collective effects in the longitudinal plane Numerical and computational tools Linear synchrotron motion: Dynamics: 17.6e10 protons, 1e12 electrons Tune footprint: Kevin Li Synchro-Betatron Motion 7/ 70

9 SPS ecloud effects Collective effects in the longitudinal plane Numerical and computational tools Nonlinear synchrotron motion: Dynamics: 17.6e10 protons, 1e12 electrons Tune footprint: Kevin Li Synchro-Betatron Motion 7/ 70

10 1 Collective effects in the longitudinal plane Numerical and computational tools Synchrotron motion does not preserve the longitudinal position over several turns The tune footprint is obtained over several turns The color dimension looses its meaning We need to find a quantity that is preserved under synchrotron motion to refurnish the color dimension with a meaning Kevin Li Synchro-Betatron Motion 8/ 70

11 1 Collective effects in the longitudinal plane Numerical and computational tools Synchrotron motion does not preserve the longitudinal position over several turns The tune footprint is obtained over several turns The color dimension looses its meaning We need to find a quantity that is preserved under synchrotron motion to refurnish the color dimension with a meaning Kevin Li Synchro-Betatron Motion 8/ 70

12 1 Collective effects in the longitudinal plane Numerical and computational tools Kevin Li Synchro-Betatron Motion 8/ 70

13 1 G. Rumolo

14 1 G. Rumolo But where does this derive from?

15 2 Collective effects in the longitudinal plane Numerical and computational tools Kevin Li Synchro-Betatron Motion 10/ 70

18 Goal Collective effects in the longitudinal plane Numerical and computational tools What this presentation should be about: not a presentation of new results by no means any claim for mathematical rigor rather an attempt to gather different ressources to summarize the known theory in a more or less complete and comprehensible manner rather with an appeal to physical intuition Kevin Li Synchro-Betatron Motion 12/ 70

19 Goal Collective effects in the longitudinal plane Numerical and computational tools What this presentation should be about: not a presentation of new results by no means any claim for mathematical rigor rather an attempt to gather different ressources to summarize the known theory in a more or less complete and comprehensible manner rather with an appeal to physical intuition Kevin Li Synchro-Betatron Motion 12/ 70

20 Literature Collective effects in the longitudinal plane Numerical and computational tools Physics: [Goldstein: Classical Mechanics] [Jackson: Classical Electrodynamics] [Huang: Statistical Mechanics] [Peskin/Schroeder: Quantum Field Theory] Applied Hamiltonian dynamics: [T. Suzuki: 1985] [K. Symon: 1997] [S. Tzenov: 2001] Kevin Li Synchro-Betatron Motion 13/ 70

21 Outline Basic physics Specialisation to classical electromagnetic theory 1 Collective effects in the longitudinal plane Numerical and computational tools in accelerator physics 2 Basic physics Specialisation to classical electromagnetic theory 3 The symplectic structure Kevin Li Synchro-Betatron Motion 14/ 70

22 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Initialisation: Basic objects Parameter manifold: Configuration manifold: Map: M = R m N = R n Φ : M N Derived objects World bubble: Phase space: Jacobian: Θ = Φ(U M) = R m Ω = T N = R 2n J = DΦ M(m n, R) All physics is in finding Φ Kevin Li Synchro-Betatron Motion 15/ 70

23 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Initialisation: We create our universe by declaring two manifolds and connecting them with a map The parameter manifold M is our world The configuration manifold N is some quantity we are interested in The map Φ is an embedding of our world into the target space and as such describes the evolution of the target space quantities Kevin Li Synchro-Betatron Motion 15/ 70

24 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Why do we need manifolds and all that stuff? In our intuition we are (always) using them We (always) start with a collection of points which are, a priori, completely unstructured (i.e. a mesh of an accelerator structure (without connectivity information) or the time steps in a particle tracking code (with no ordering)) We want to be able to talk about neighbourhoods, derivatives, tangent spaces, metrics in order obtain a predictable evolution (a function of the parameter manifold) for any quantity that lives in our world. Our collection of points must thus be endowed with a smooth connectivity which is done formally via a differentiable structure (equivalence class of atlases where an atlas is a family of compatible charts on an open cover of the parameter manifold 1 ). Then, locally, our collection of points becomes isomorph to the Euclidean space; it locally obtains the structure of a linear vector space within which we are fully equipped with all our well-known tools of calculus 1 s. Abraham, Marsden pp. 31 Kevin Li Synchro-Betatron Motion 15/ 70

25 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Why do we need manifolds and all that stuff? In our intuition we are (always) using them We (always) start with a collection of points which are, a priori, completely unstructured (i.e. a mesh of an accelerator structure (without connectivity information) or the time steps in a particle tracking code (with no ordering)) We want to be able to talk about neighbourhoods, derivatives, tangent spaces, metrics in order obtain a predictable evolution (a function of the parameter manifold) for any quantity that lives in our world. Our collection of points must thus be endowed with a smooth connectivity which is done formally via a differentiable structure (equivalence class of atlases where an atlas is a family of compatible charts on an open cover of the parameter manifold 1 ). Then, locally, our collection of points becomes isomorph to the Euclidean space; it locally obtains the structure of a linear vector space within which we are fully equipped with all our well-known tools of calculus 1 s. Abraham, Marsden pp. 31 Kevin Li Synchro-Betatron Motion 15/ 70

26 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Why do we need manifolds and all that stuff? In our intuition we are (always) using them We (always) start with a collection of points which are, a priori, completely unstructured (i.e. a mesh of an accelerator structure (without connectivity information) or the time steps in a particle tracking code (with no ordering)) We want to be able to talk about neighbourhoods, derivatives, tangent spaces, metrics in order obtain a predictable evolution (a function of the parameter manifold) for any quantity that lives in our world. Our collection of points must thus be endowed with a smooth connectivity which is done formally via a differentiable structure (equivalence class of atlases where an atlas is a family of compatible charts on an open cover of the parameter manifold 1 ). Then, locally, our collection of points becomes isomorph to the Euclidean space; it locally obtains the structure of a linear vector space within which we are fully equipped with all our well-known tools of calculus 1 s. Abraham, Marsden pp. 31 Kevin Li Synchro-Betatron Motion 15/ 70

27 Basic physics Specialisation to classical electromagnetic theory We create our universe: two manifolds and one map Why do we need manifolds and all that stuff? In our intuition we are (always) using them We (always) start with a collection of points which are, a priori, completely unstructured (i.e. a mesh of an accelerator structure (without connectivity information) or the time steps in a particle tracking code (with no ordering)) We want to be able to talk about neighbourhoods, derivatives, tangent spaces, metrics in order obtain a predictable evolution (a function of the parameter manifold) for any quantity that lives in our world. Our collection of points must thus be endowed with a smooth connectivity which is done formally via a differentiable structure (equivalence class of atlases where an atlas is a family of compatible charts on an open cover of the parameter manifold 1 ). Then, locally, our collection of points becomes isomorph to the Euclidean space; it locally obtains the structure of a linear vector space within which we are fully equipped with all our well-known tools of calculus 1 s. Abraham, Marsden pp. 31 Kevin Li Synchro-Betatron Motion 15/ 70

28 Basic physics Specialisation to classical electromagnetic theory Basic physics: the principle of least action In this case we can define: Action The action S is defined as the volume of the world bubble: S = dφ The principle of least action Given a fixed subspace U, a map Φ is physical if and only if the action S is stationary δs = 0 Θ Kevin Li Synchro-Betatron Motion 16/ 70

29 The Lagrangian Basic physics Specialisation to classical electromagnetic theory To introduce a useful formalism it is expedient to write the action as Action S = Φ(U) dφ = U d m x det(j J T ) = d m x L U Thus, we have introduced the Lagrangian Lagrangian L = det(j J T ) Kevin Li Synchro-Betatron Motion 17/ 70

30 Basic physics Specialisation to classical electromagnetic theory Classical limit and electromagnetic theory Introduce classical limit δ = nλ 3 1 δ: characteristic dimensionless density parameter of a quantum gas λ = : thermal de Broglie wavelength 2π 2 mk B T The action becomes the length of the world-line: S = dl, dl 2 = c 2 dt 2 + d x 2 Introduce electromagnetic theory via U(1)-gauge coupling by moving from the standard to the covariant derivative A: gauge fields µ D µ = µ + iga µ The action becomes the covariant length of the world-line: S = D µl = L dt Kevin Li Synchro-Betatron Motion 18/ 70

31 Intuition of the classical limit Basic physics Specialisation to classical electromagnetic theory A small number of particles with a wavefunction that ( is represented ) as an evolving Gaussian wavepackage: ψ(x, t) exp x2 σ 2 Kevin Li Synchro-Betatron Motion 19/ 70

Intuition of the classical limit Basic physics Specialisation to classical electromagnetic theory For sufficiently many particles at low densty constructive

32 Intuition of the classical limit Basic physics Specialisation to classical electromagnetic theory For sufficiently many particles at low densty constructive superposition of wavefunctions establishes a correlation between space and time coordinates via delta-functions: ψ(x, t) δ(x(t) x ) x(t) Kevin Li Synchro-Betatron Motion 19/ 70

33 Intuition of electromagnetic theory Basic physics Specialisation to classical electromagnetic theory Gauge invariance of the Dirac field Directional derivative ψ(x) e iα(x) ψ(x) e ψ( x) = lim h 0 ψ( x + h e) ψ( x) h Kevin Li Synchro-Betatron Motion 20/ 70

34 Intuition of electromagnetic theory Basic physics Specialisation to classical electromagnetic theory Gauge invariance of the Dirac field Directional derivative ψ(x) e iα(x) ψ(x) e ψ( x) = lim h 0 ψ( x + h e) ψ( x) h ψ( x) µ ψ( x) Kevin Li Synchro-Betatron Motion 20/ 70

derivative ψ(x) e iα(x) ψ(x) e ψ( x) = lim h 0 ψ( x + h e) ψ( x) h e iα(x) ψ(

35 Intuition of electromagnetic theory Basic physics Specialisation to classical electromagnetic theory Gauge invariance of the Dirac field Directional derivative ψ(x) e iα(x) ψ(x) e ψ( x) = lim h 0 ψ( x + h e) ψ( x) h e iα(x) ψ( x) µ ψ( x) ψ( x + h e) D µ = ( µ + iea µ) ψ( x) Kevin Li Synchro-Betatron Motion 20/ 70

36 Basic physics Specialisation to classical electromagnetic theory One-parameter action and electromagnetic Lagrangian One-parameter action and electromagnetic Lagrangian S = L d 4 x S = L = ρ m c ẋ µ ẋ µ + j µ A µ L = mc L dt 1 v2 qv + q v A c2 Kevin Li Synchro-Betatron Motion 21/ 70

37 Electromagnetic Hamiltonian Basic physics Specialisation to classical electromagnetic theory Hamiltonian A Legendre transform of the Lagrangian yields the Hamiltonian H = P q L with P = L q H(q, P, t) = ( P q A) 2 c 2 + m 2 c 4 + qv (1) Kevin Li Synchro-Betatron Motion 22/ 70

38 Outline Symplectic structure 1 Collective effects in the longitudinal plane Numerical and computational tools in accelerator physics 2 Basic physics Specialisation to classical electromagnetic theory 3 The symplectic structure Kevin Li Synchro-Betatron Motion 23/ 70

39 Symplectic structure Least action and Hamilton equations of motion δs = δ P dq H dt = δp dq + P δ(dq) δh dt H δ(dt) = dq δp dp δq H H H dt δq dt δp = = 0 P.I. ( dq H ) P dt q δp P ( dp + H q dt ) δq + dt δt + dh δt ) δt t ( dh H t dt Equations of motion q = H P, P = H q, Ḣ = H t Kevin Li Synchro-Betatron Motion 24/ 70

40 Symplectic structure Symplectic structure The Legendre transform makes the independent variable time and together with the principle of least action/equations of motion unleashes the full symplectic structure of the theory yielding: the symplectic manifold (Ω, ω 0 ) (Phase space, Poisson bracket) ω 0 : T q N T q N R, (u, v) ω 0 (u, v) because ω 0 is nondegenerate, it defines a 1-form ω 1 : T q N T q N, u ω 0 (u, ) ( u ) let s use this 1-form to implicitly define a very special vector field ω 0 (X H, Y ) = dh(y ) (JX H, Y ) = ( H, Y ) q N, J = ( ) 0 1 symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 25/ 70

41 Symplectic structure Symplectic structure The Legendre transform makes the independent variable time and together with the principle of least action/equations of motion unleashes the full symplectic structure of the theory yielding: the symplectic manifold (Ω, ω 0 ) (Phase space, Poisson bracket) ω 0 : T q N T q N R, (u, v) ω 0 (u, v) because ω 0 is nondegenerate, it defines a 1-form ω 1 : T q N T q N, u ω 0 (u, ) ( u ) let s use this 1-form to implicitly define a very special vector field ω 0 (X H, Y ) = dh(y ) (JX H, Y ) = ( H, Y ) q N, J = ( ) 0 1 symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 25/ 70

42 Symplectic structure Symplectic structure The Legendre transform makes the independent variable time and together with the principle of least action/equations of motion unleashes the full symplectic structure of the theory yielding: the symplectic manifold (Ω, ω 0 ) (Phase space, Poisson bracket) ω 0 : T q N T q N R, (u, v) ω 0 (u, v) because ω 0 is nondegenerate, it defines a 1-form ω 1 : T q N T q N, u ω 0 (u, ) ( u ) let s use this 1-form to implicitly define a very special vector field ω 0 (X H, Y ) = dh(y ) (JX H, Y ) = ( H, Y ) q N, J = ( ) 0 1 symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 25/ 70

43 Symplectic structure Symplectic structure The Legendre transform makes the independent variable time and together with the principle of least action/equations of motion unleashes the full symplectic structure of the theory yielding: the symplectic manifold (Ω, ω 0 ) (Phase space, Poisson bracket) ω 0 : T q N T q N R, (u, v) ω 0 (u, v) because ω 0 is nondegenerate, it defines a 1-form ω 1 : T q N T q N, u ω 0 (u, ) ( u ) let s use this 1-form to implicitly define a very special vector field ω 0 (X H, Y ) = dh(y ) (JX H, Y ) = ( H, Y ) q N, J = ( ) 0 1 symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 25/ 70

44 Hamiltonian vector field Symplectic structure We have thus defined the Hamiltonian vector field X H = J H =: H : What is so special about this Hamiltonian vector field? Infinitesimal time evolution Finite time evolution X H = J H = f α=1 H q α H p α p α q α ψ(t 0) = X H ψ(t 0) = : H : ψ(t 0) = [H, ψ(t 0)] ψ(t 0 + t) = exp( : H : t)ψ(t 0) The Hamiltonian ( is the ) generator for translations in time for any function ψ! 0 1 q N, J = symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 26/ 70

45 Hamiltonian vector field Symplectic structure We have thus defined the Hamiltonian vector field X H = J H =: H : What is so special about this Hamiltonian vector field? Infinitesimal time evolution Finite time evolution X H = J H = f α=1 H q α H p α p α q α ψ(t 0) = X H ψ(t 0) = : H : ψ(t 0) = [H, ψ(t 0)] ψ(t 0 + t) = exp( : H : t)ψ(t 0) The Hamiltonian ( is the ) generator for translations in time for any function ψ! 0 1 q N, J = symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 26/ 70

46 Hamiltonian vector field Symplectic structure We have thus defined the Hamiltonian vector field X H = J H =: H : What is so special about this Hamiltonian vector field? Infinitesimal time evolution Finite time evolution X H = J H = f α=1 H q α H p α p α q α ψ(t 0) = X H ψ(t 0) = : H : ψ(t 0) = [H, ψ(t 0)] ψ(t 0 + t) = exp( : H : t)ψ(t 0) The Hamiltonian ( is the ) generator for translations in time for any function ψ! 0 1 q N, J = symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 26/ 70

47 Hamiltonian vector field Symplectic structure We have thus defined the Hamiltonian vector field X H = J H =: H : What is so special about this Hamiltonian vector field? Infinitesimal time evolution Finite time evolution X H = J H = f α=1 H q α H p α p α q α ψ(t 0) = X H ψ(t 0) = : H : ψ(t 0) = [H, ψ(t 0)] ψ(t 0 + t) = exp( : H : t)ψ(t 0) The Hamiltonian ( is the ) generator for translations in time for any function ψ! 0 1 q N, J = symplectic structure matrix 1 0 Kevin Li Synchro-Betatron Motion 26/ 70

48 Liouville s theorem Symplectic structure One of the many corollaries: preservation of the volume form on Ω (Liouville s theorem) Kevin Li Synchro-Betatron Motion 27/ 70

49 Liouville s theorem Symplectic structure One of the many corollaries: preservation of the volume form on Ω (Liouville s theorem) Kevin Li Synchro-Betatron Motion 27/ 70

50 Liouville s theorem Symplectic structure One of the many corollaries: preservation of the volume form on Ω (Liouville s theorem) Kevin Li Synchro-Betatron Motion 27/ 70

51 Symplectic structure Liouville s theorem One of the many corollaries: preservation of the volume form on Ω (Liouville s theorem) Kevin Li Synchro-Betatron Motion 27/ 70

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