Outline. Liouville s theorem Canonical coordinates Hamiltonian Symplecticity Symplectic integration Symplectification algorithm

Transcription

1 Outline Liouville s theorem Canonical coordinates Hamiltonian Symlecticity Symlectic integration Symlectification algorithm USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

2 Liouville s Theorem In the local region of a article, the article density in hase sace is constant rovided that the articles move in a general field consisting of magnetic fields and of fields whose forces are indeendent of velocity. Density function = number of articles er unit volume of six-dimensional hase sace Particle current f ( r,, t) J6 ( fr, f ) ( f, ff) Continuity equation = reservation of the number of articles f / t 6 J 6 6J6 ( f) ( f F) ( f) f ( ) ( f) F f( F) F [ gr ( ) q Br ( )] q ( B ) qb ( ) q ( B ) y x c mc x c mc y c mc z USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

3 Continuity equation becomes Theorem roved. Liouville s Theorem 6J6 ( f ) ( f ) F f / t( f) r ( f) df / dt Limitations and excetions Linear and non-linear mismatch effective emittance growth Velocity-deendent Coulomb interaction between individual articles: IBS, Touschek Dissiative mechanisms Radiation, radiation daming Charge-exchange inection Electron cooling Stochastic cooling An immediate consequence of the theorem T T Y T( X), T () X Y () X M X i, i i 1X 1X 1 Since the determinant of the Jacobian is a ratio of volumes det M 1 USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

4 Canonical Momentum and Vector Potential Conservative force Lorentz force F F d / dt q( E B) can be velocity deendent and is not always conservative. F d/ dt q[ E( B)] qb / tq[( B ) ( ) B( B) ( ) B] qb/ t q( ) B qdb / dt d( qa)/ dt d d d ( qa) ( qa) dt dt dt Canonical momentum and conservative canonical force P qa, F dp/ dt can USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

5 Relativistic Hamiltonian of a free article Hamiltonian in an electromagnetic field Hamilton s equations Frenet-Serret curvilinear coordinate system Hamiltonian becomes Hamiltonian H c m c 4 B A, E A/ t 4 H ( PqA) c m c q dp / dt H, dx / dt PH ˆ ˆ ˆ ˆ r x dl dr r rd dy y dx x ds sˆdy yˆ dx xˆdy yˆ(1 ) ds sˆ 4 s H m c c x y 1 x / H Ps qas c m c ( Px qax) ( Py qay) q 1 x / USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

6 s-hamiltonian Poincaré-Cartan invariant under a canonical variable transformation Hamiltonian with the longitudinal coordinate as the indeendent variable Curl in Frenet-Serret coordinates For static transverse magnetic fields Pdr Hdt inv PdxPdyPdsHdt PdxPdy( H) dt( P) ds x y s x y s PdxPdy( U) dth ds x y s H P( x, P, y, P, t, U; s) s s x y x U q s 1 ( x x) ( y y) qa m c P qa P qa c 1 A A s y 1 Ax A A s y Ax A xˆ yˆ sˆ 1 x/ y s 1 x/ s x x y B x, A A x 1 A s 1, B A s y 1 x/ y 1 x/ x y USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

7 Hamiltonian in Standard Canonical Coordinates For static transverse magnetic fields Canonical change of longitudinal coordinates, A A x y H s qas x U mc x y 1 c qas x U mc 1 x y c H( x, x, y, y, t, U / ; s) / dx/ ds x, / dy/ ds y x y ( t, U / ) ( z s t, / ) H ( x, x, y, y, z, ; s) H( x, x, y, y, t, U / ; s) F ( t, ; s)/ s z F /, U / F / t USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

8 Hamiltonian in Standard Canonical Coordinates U F tf(, s) U U U c 1 (1 ) U Hamiltonian in the new canonical longitudinal coordinates c F (1 ) tsf( s) c s s F (1 ) t s qas x H 1 x y qa x F s s 1 1 x y 1 USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

9 Hamiltonian in Standard Canonical Coordinates Exansion of the Hamiltonian keeing the lower-order terms qas x 1 1 H 1 1 [ ( x y ) ] [4 ] 1 8 qas x ( x y ) 1 qas 1 ( x x x y ) USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

10 Symlectic Transformations and Matrices Hamilton s equations H H( xp,, yp,, t, Us ; ), X( xp,, yp,, t, U) x y x y dxi H dpi H, ds P ds x dx ds i i 6 1 S i H X S i USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

11 Symlectic Transformations and Matrices Canonical coordinate transformation in assing through a beam line X X ( X ) 1 1 dx dx dx 1i X M ds X ds ds 1i i dx 1i H H X 1 T k H Si Si Si M ds X1, k 1, k k X k X X k T dx M Skl dx ds 1 l k dx dx 1 T l M S M 1i Si kl i k, k ds ds l ds M SM T 1 T ( ) S, S M S M Definition of a symlectic matrix. USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

12 Symlectic Integration Exanding the equation of motion in the neighborhood of the reference traectory ˆ () X s d ˆ H ˆ H ( ) ( ) ( ˆ H X ) ( ˆ i Xi Si X X Si X X) Xk ds 1 X 1 X k1x X k dxi H S ( ˆ i X) Xk SiCkXk ds 1 k1 X X k 1 k1 C k In matrix notation If C is constant dx SC X ds SCs X () s e X() M() s e SCs If C is a function of s we aroximate it in a iecewise constant fasion as M() s e e e e SC( sds) ds SC( ds) ds SC( ds) ds SC() ds USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

13 Symlectic Matrices Since any nn symlectic matrix can be exressed in terms of a symmetric matrix it has the same number of degrees of freedom ( n) E.g. a 44 symlectic matrix has 1 indeendent arameters. Choosing a basis for 44 symmetric matrices c n n(n1) n :,,,, ,,,, , USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

14 Symlectic Matrices Any 44 symmetric matrix can be written in the form Any 44 symlectic matrix can be written as C 1 c exscex Sc ex G 1 1 G USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

15 Maniulations with Exonentials The Baker-Cambell-Hausdorff (BCH) formula X Y Z e e e X Y Z log( e e ) X Y [ X, Y] ([ X, X, Y] [ Y, Y, X]) [ X, Y, Y, X] 1 4 [ X, XY, ] [ X,[ XY, ]], [ XYYX,,, ] [ X,[ Y,[ YX, ]]] The Zassenhaus formula U to nd order in h 3 4 e e e e e e ( AB) h Ah Bh [ ABh, ] / ([ BAB,, ] [ A, AB, ]) h / 6 O( h ) e e e e O( h ) ( AB) h Ah/ Bh Ah/ SCh M( h) e ex Gh ex1g1h Gh 1 1 ex / ex e Gh G h x Gh/ half-ste drift half-ste drift thin element kick USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

16 Symlectic Integrator The matrix reresenting a canonical coordinate transformation over a small ste during which the fields can be considered constant can be reresented as a thin kick sandwiched between two drifts with all coefficients determined by the Hamiltonian G M ( h) ex 1 1 / ex ex 1 1 / half-ste drift half-ste drift thin element kick Gh Gh Gh Sc h / 1 3 ex( 1Gh 1 / ) O( h ) 1 1h / 1 This is the basis of symlectic drift-kick codes, kicks can be non-linear, e.g. for sextuoles, octuoles, etc. USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

17 Time-Symmetric Integrators Aly the BCH formula to the time symmetric roduct Xh Y e e e e ex( X Y [ XY, ] ([ X, XY, ] [ YY,, X]) [ XYY,,, X]) e ex{( X Y) h [ X, Yh ] ([ X, XY, ] [ YYX,, ]) h [ XYYXh,,, ] Xh [ XYXh, ] [[ XY, ], Xh ] ([[ X, XY, ], X] [ [ YYX,, ], X]) h ([ X Y, X Y, X] [ X, X, X Y]) h [ X, X, X, Y] h Oh ( )} ex{( X Y ) h ([ YY,, X ] [ X, XY, ]) h O( h )} 6 W h Xh Xh USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

18 If an integrator is time reversible then Alying the BCH formula gives Time-Symmetric Integrators Ih ( ) e Reeating we find by induction that all even owers vanish log( Ih ( )) i.e. must be an odd function of h ghg h gh IhI ( ) ( h) IhIhe e e gh g h gh gh g h gh gh Oh ( ) ( ) ( ) 1 g g USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

19 Higher-Order Integrators Higher-order integrators may be constructed from the nd -order integration function 4 th -order integrator may be constructed using time-symmetric roduct of nd -order integrators To 4 th -order this requires Ah/ Bh Ah/ gh 1 gh 3 I ( h) e e e e I ( h) I ( ah) I ( bh) I ( ah) ( ) a b g h a b g h A B h O h I4 h e e ab a b ( ) ( ) ( ) ( ) 1/3 1 a, b 1/3 1/3 USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

20 Symlectification Algorithm A symlectic matrix M may be written in the form if and only if W is a symmetric matrix. If W is symmetric If M is symlectic and M ( I SW)( I SW) M SM ( I W S ) ( I W S ) S( I SW)( I SW) T T T 1 T T 1 ( I WS) ( I WS) S( I SW )( I SW ) ( I WS) ( I WS)( I WS) S( I SW ) ( I WS) ( I WS)( I WS) S( I SW ) ( I WS) ( I WS) S( I SW )( I SW ) S T W PQ, W P Q M SM S ( I W S ) ( I W S ) S( I SW)( I SW) T T T 1 T T 1 ( I W S ) ( S PQPQW SW)( I SW) T T 1 T 1 ( I W S ) [( S W )( I SW ) 4 Q]( I SW ) T T 1 T 1 S I W S Q I SW Q T T 1 1 4( ) ( ) T 1 USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

21 Symlectification Algorithm Symmetric matrix from a symlectic matrix For an almost symlectic matrix calculate the almost symmetric matrix Symmetrize the almost symmetric matrix W S I M I M 1 ( ) ( ) W S( I M ) ( I M ) 1 T 1 ( W W )/ Obtain a symlectic aroximation of the original matrix W M ( I SW )( I SW ) USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January

22 Sector diole magnet: Hessian matrix of the Hamiltonian Examle: Sector Diole Magnet B yˆ, for s L B, elsewhere As x s 1 By A B x 1 x/ x x x 1 x x x 1 x H ( x y ) ( x y ) 1/ / c1 ( 3 4), SC H C c c X ix 1 1 cos( s/ ) sin( s/ ) n SCs ( SCs) sin( s/ ) / cos( s/ ) M() s e n n! 1 s 1 USPAS: Lecture on Hamiltonian Dynamics Vasiliy Morozov, January