Symplectic Maps for General Hamiltonians

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1 Symplectic Maps for General Hamiltonians Malte Titze November 2, 2015

2 Outline 1 Introduction 2 Canonical Transformations 3 Obtaining the Map M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 2 / 21

3 Motivation In order to track a system over a long period of time, it must be numerically robust. For an overview see Oeftiger (2014). Symplectic integrators aim to solve the differential equations (locally) exact: For every step size, they provide a map transporting the initial coordinates to the final ones, while respecting the Hamilton equations. In particular, I began to work on a thin-lens implementation of combined function magnets into MADX, in order to use them for the frozen-sc model in the PS: We derived symplectic kick matrices, which are sufficient for twiss, see Titze (2015). However, the underlying first-order 1 map without the drifts described there does not have to be symplectic (Reason: truncation). But we also require symplectic maps for tracking. 1 in path-length M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 3 / 21

4 Main Idea Combine Ripken s ideas (Barber et al. (1996)) with the ones we used at HZB 2 to track symplectic through thick multipoles (Titze et al. (2015)). Use Taylor expansion of a generating function around the final position (i.e. work locally). Deal with the higher time- resp. s-derivatives. Automatically obtain implicit symplectic formulas for the final coordinates in dependency of the original ones. 2 Helmholtz-Zentrum Berlin M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 4 / 21

5 Outline 1 Introduction 2 Canonical Transformations 3 Obtaining the Map M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 5 / 21

6 Hamilton-equations revisited, p. I Let U R 2n+1 be an open subset of the space of the canonical variables q, p R n. Denote by s : U R the projection onto the last component. For a given Hamiltonian H on U consider the 2-form ω H := dp dq dh ds Ω 2 (U). We observe: ω H : X(U) Ω 1 (U), X ω H (X ), defines a 1-dimensional distribution on the tangent bundle TU via its kernel, which is horizontal with respect to the 2n-bundle s : U s(u): Proof. If X = α s + A i q i + B i p i X(U) is a vector field, then the condition ω H (X ) = 0 leads to A i = α p i H, B i = α q i H. Therefore: α = 0 X = 0, i.e. the kernel is 1-dimensional. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 6 / 21

7 Hamilton-equations revisited, p. II From the proof we see: Integral curves of X with ω H (X ) = 0 and α 1 obey the Hamilton-equations. Why this reformulation? We will see that the theory of generating functions and its connection to symplectic transformations fits naturally into this framework. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 7 / 21

8 Symplectic Transformations Let G : U U be a diffeomorphism which is the identity on the last component, and K another Hamiltonian. Denote by G =: ( q, p, s) the representation of G wrt. the first- and second n components. We say G is canonical (or symplectic) trf. from H to K : G ω K = ω H. Since G dq = d q and G dp = d p, this is equivalent to: d p d q d(k G) ds = dp dq dh ds Consequence: canonical G Hamilton-eqs. wrt. H Hamilton-eqs. wrt. K G. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 8 / 21

9 Generating Functions Previous equation G ω K = ω H again: Observe: dp dq d p d q + d(k G H) ds = 0. d[(k G H)ds] = d(k G H) ds. d(p k dq k ) = d( q k dp k ) = dp k dq k and similar for d p k d q k. In total 2n of these terms, which leads to 2 2n different 1-forms α s with dα s = 0. Poincaré s (locally) generating function F s with df s = α s. Example (which we will use in this talk): α := p k dq k + q k d p k + (K ψ H ϕ)ds, with isomorphisms ψ(q, p, s) := ( q, p, s) and ϕ(q, p, s) := (q, p, s). F with df = α satisfies: q F = p, p F = q and s F = K ψ H ϕ. if F is known, then G implicitly. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 9 / 21

10 Assumptions Since we are working locally, we can assume 1 We can unfold the distribution ω H to a trivial one ω 0 : Its kernel has no components in vertical direction. 2 This transformation is the identity at a certain position s f. In other words: 1 K 0 q 0 and p 0, i.e. there exist a symplectic transformation into cyclic (comoving) coordinates. 2 This system is parameterized by the coordinates q f, p f at the position s f. Previous slide: the resulting map from (q, p, s) to (q f, p f, s f ) is automatically symplectic. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 10 / 21

11 Outline 1 Introduction 2 Canonical Transformations 3 Obtaining the Map M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 11 / 21

12 By the first assumption, F must satisfy: q F = p, p F = q, s F = H ϕ, (1a) (1b) (1c) with ϕ(q, p, s) = (q, p, s). Let us assume that s := s f s is small and consider the Taylor-expansion (K N 0 { }) K+1 1 F (q, p, s) = F 0 (q, p) + µ! ( s)µ s µ F (q, p, s f ) (2) µ=1 The second assumption is fulfilled if we set F 0 = q p: Insert F 0 into (1a), (1b) yield p = p f and q = q f. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 12 / 21

13 We now use (1c) to express the higher derivatives of F in terms of H: F (q, p, s) = q p K µ=0 How to deal with the unknown Function ϕ? Lemma 1 (proof see Appendix) 1 (µ + 1)! ( s)µ+1 µ s (H ϕ)(q, p, s f ). Let F, ϕ as above and G a differentiable function of (q, p, s). Then q i (G ϕ) = [ q i G + ( p j G) ( q i p j ϕ 1 ) ] ϕ. p i (G ϕ) = [( p j G) ( p i p j ϕ 1 )] ϕ. (3a) (3b) s (G ϕ) = [ p i G q i H p i G ( q i p j ϕ 1 ) p j H + s G ] ϕ. (3c) M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 13 / 21

14 For brevity let us write S ij := q i p j ϕ 1. Then Lemma 1 yield: s (S ij ϕ) = [ p l S ij q l H p l S ij S lk p k H + s S ij ] ϕ, and s (S ij ϕ) = s ( q i p j ) = s q i q j F = q i q j (H ϕ) ( = q i [ q j H + p l H S jl ] ϕ ) Consequently = ( q i [ q j H + p l H S jl ]+ + p k [ q j H + p l H S jl ] S ik ) ϕ s S ij = q i [ q j H + p l H S jl ] p k [ q j H + p l H S jl ] S ik + + p l S ij q l H + p k S ij S kl p l H. (4) This equation shows that the s-derivatives of S ij of any order can entirely be expressed by their spacial derivatives and derivatives of the Hamiltonian, with the Hesse-matrix of H as generator. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 14 / 21

15 By the Taylor-expansion of F : p = q F = p K µ=0 1 (µ + 1)! ( s)µ+1 q µ s (H ϕ)(q, p; s f ) (5) so all derivatives of q i p j with respect to q and p vanish at s = s f. This means that S ij s f 0 and therefore also all its higher spacial derivatives vanish at s f. Denote therefore with (3c): H.G := p i G q i H + p i G S ij p j H s µ (G ϕ) = s µ 1 (( H. + s )G ϕ) = = ( H. + s ) µ G ϕ and so we obtain F (q, p; s) = q p K µ=0 1 (µ + 1)! ( s)µ+1 (( H. + s ) µ H)(ϕ(q, p; s f )) M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 15 / 21

16 The Symplectic Map Since q i (G ϕ)(q, p; s f ) = ( q i G ϕ)(q, p; s f ) and p i (G ϕ)(q, p; s f ) = ( p j G ϕ)(q, p; s f ) p i p j s f by Lemma 1, }{{} we obtain the expressions δ ij q = q p = p + K µ=0 K µ=0 1 (µ + 1)! ( s)µ+1 ( p ( H. + s ) µ H)(q, p; s f ), (6a) 1 (µ + 1)! ( s)µ+1 ( q ( H. + s ) µ H)(q, p; s f ), (6b) using that ϕ(q, p; s f ) = (q, p(q, p; s f ); s f ) = (q, p; s f ). Implicit symplectic transformation for every K: Equation (6b) can be solved analytically or - in more complicated cases - numerically. Then insert its result into (6a). M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 16 / 21

17 Summary 1 We have shown how to obtain (local) symplectic transformations for general Hamiltonians. We emphasize that this works especially for time-dependent Hamiltonians as well as those in which there is no splitting between p and q variables possible (for example bending- and CF-magnets). 2 The transformation stays symplectic even if we truncate at a given order K. 3 A first demonstration code has been written to compute the series (6a), (6b) and obtain solutions for basic Hamiltonians like the 1D RMS envelope equation with driving term. 4 No hassle with a particular differential equation required. 5 If one series leads to a complicated implicit equation, one can try another generating function. Many thanks for your attention! M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 17 / 21

18 Further Reading D. P. Barber, K. Heinemann, G. Ripken, and F. Schmidt. Symplectic thin-lens transfer maps for sixtrack: Treatment of bending magnets in terms of the exact hamiltonian. Internal report, J. R. Cary. Lie transform perturbation theory for hamiltonian systems. Physical Letters, 79(2): , A. Oeftiger. Symplectic integrators. Space Charge WG meeting, M. Titze. Thin-lens implementation of combined-function magnets in mad-x. Computing Meeting, September URL Meetings/combined_function_ /. M. Titze, J. Bahrdt, and G. Wüstefeld. Symplectic tracking through three dimensional magnetic fields by a method of generating functions. To be published, H. Yoshida. Construction of higher order symplectic integrators. Physical Letters A, 150(5), M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 18 / 21

19 Details of first claim Let X = α s + A i q i + B i p i and ω H = dp dq dh ds. Then using dp(x ) = X (p), dq(x ) = X (q), ds(x ) = X (s) and dh(x ) = X (H): 0 = ω H (X ) = A i dp i + B i dq i + αdh dh(x )ds = A i dp i + B i dq i + α q i Hdq i + α p i Hdp i + α s Hds + A i q i Hds B i p i Hds α s Hds = ( A i + α p i H)dp i + (B i + α q i H)dq i + (A i q i H + B i p i H)ds which is equivalent to A i = α p i H, B i = α q i H. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 19 / 21

20 Proof of Lemma 1, p. I We proof Lemma 1: q i (G ϕ)(q, p, s) = ( q j G)(ϕ(q, p, s)) q i q j + ( p j G)(ϕ(q, p, s)) q i p j + ( s G)(ϕ(q, p, s)) q i s = [ ( q i G) + ( p j G) ( q i p j ϕ 1 ) ] (ϕ(q, p, s)). p i (G ϕ)(q, p, s) = ( q j G)(ϕ(q, p, s)) p i q j + ( p j G)(ϕ(q, p; s)) p i p j + ( s G)(ϕ(q, p, s)) p i s = ( p j G)(ϕ(q, p, s)) p i p j. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 20 / 21

21 Proof of Lemma 1, p. II and s (G ϕ)(q, p, s) =( q i G)(ϕ(q, p, s)) s q i + }{{} 0 ( p i G)(ϕ(q, p, s)) s p i }{{} s q i F with (1c) +( s G)(ϕ(q, p, s)) = ( p i G)(ϕ(q, p, s))( q i (H ϕ))(q, p, s)+ ( s G)(ϕ(q, p, s)) = [ p i G q i H p i G ( q i p j ϕ 1 ) p j H + s G ] (ϕ(q, p, s)), where we have used (1a) for the last equation. M. Titze (CERN / University of Berlin) Symplectic Maps for General Hamiltonians 21 / 21

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