Guiding-center transformation 1. δf = q ( c v f 0, (1) mc (v B 0) v f 0. (3)

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1 Guidig-ceter trasformatio 1 This is my otes whe readig Liu Che s book[1]. 1 Vlasov equatio The liearized Vlasov equatio is [ t + v x + q m E + v B c ] δf = q v m δe + v δb c v f, 1 where f ad δf are the equilibrium ad perturbed distributio fuctios, respcetively, E, B, δe, ad δb are the equilibrim ad perturbed electromagetic field. We cosider the case of E =. Defie the uperturbed Vlasov propagator the Eq. 1 is writte as F t +v x + q mc v B v, 2 Fδf = q m δe + v δb c v f. 3 2 Guidig-ceter trasformatio We ow cosider the guidig-ceter trasformatio i uiform magetic field. The trasformatio from the particle variables x, v to the guidig-ceter variables X, ε, µ, α, is defied as X = x+v e Ω, 4 ε= v2 2, 5 µ = 2 /2B, 6 = sgv, 7 ad α is the gyrophase agle which is defied i the followig. Here e = B /B, Ω = qb /mc. I terms of ε, µ, α,, the parallel ad perpedicular velocity of a particle are give respectively by v = 2ε B µ, 8 ad = 2B µ e 1 cosα +e 2 siα, 9 where e 1 ad e 2 are two orthogoal uit vectors perpedicular to B ; e 1 e = ad e 2 = e e 1, thus Eq. 9 idicates the gyrophase agle is defied as the icluded agle betwee ad e 1. We ow cosider the trasformatio of the /x ad /v operators to the guidig-ceter variables. For otatio coveiece, we defie V =ε, µ, α,. The we have From Eq. 4, we obtai x = X x X + V x V 1 X = I, 11 x ad, sice the equilibrium magetic field is uiform, the defiitio of ε, µ, α, ad do ot ivolve spatial variables, thus V is idepedet of x, i.e., V =. 12 x

2 2 Sectio 2 Usig the above results i Eq. 1 gives x = I X + V = X. 13 Now cosider the trasformatio of the gradiet i velocity space, /v, From Eq. 4, we get v = X v X + V v V X v = v e v Ω = 1 v e Ω v The secod term of r.h.s. of Eq. 14 ca be writte as Usig ad Eq. 16 is writte as where V v V v V = ε v α v = 1 Usig Eqs. 15 ad 2 i Eq. 14 yields v = I e Ω 14 = 1 Ω I e 15 ε + µ v µ + α v α 16 ε =v, 17 v µ v =, 18 B e V = v ε +v B µ + e α 19 α, 2 e α =e. 21 X + v ε + v B µ + e α Usig Eqs. 13 ad 22, the uperturbed Vlasov propagator ca be writte, term by term, as q mc v B v α 22 F t +v x + q mc v B v, 23 v t = t x = v X = I e = v Ω = v Ω 24 + v X 25 X Ω X +v ε + v B µ + e α α I e Ω X + e α α = v e [ I e = Ω X α X ] + v Ω eα α 26

3 Solutio to the liearized equatio i the electrostatic limit 3 Usig Eqs. 24, 25 ad 26, the uperturbed Vlasov propagator, F, is writte as The equilibrium equatio F = t + v Ω X α F g. 27 F g f g =, 28 the reduces to sice equilibrium distributio fuctio is idepedet of time ad spatial locatio Ω α f g =, 29 Here, the subscript g stadig for guidig-ceter deotes a quatity of guidig-ceer variables, X, V. The solutio to Eq. 29 is obvious, i.e., f g = f g ε, µ,, 3 or equivaletly, i term of the usual coordiators, the equilibrium distributio fuctio is writte as f = f, v. 31 The liearized Vlasov equatio i guidig-ceter coordiates is writte as t +v Ω δf g = q δe + v δb X α m c v f = q δe g + v δb g v m c ε + v B µ where use has bee made of that f is idepedet of X ad α. f g, 32 3 Solutio to the liearized equatio i the electrostatic limit I the electrostatic limit, we have δe = δφ x 33 = δφ g X 34 the the liearized Vlasov equatio i guidig-ceter coordiators is writte as F g δf g = q δφ g m X v ε + v f g ε, µ,, B µ 35 which ca be arraged ito the form Notig that F g δf g = q m δφ g X [ v ε + ε + ] f g 36 B µ δφ v =, 37 Trasformig to guidig-ceter coordiates, Eq. 37 is writte as I e Ω X +v ε + v B µ + e α δφ g =. 38 α Dottig the above equatio by e α, ad otig that e α = ad e α v =, we obtai φ g X = Ω δφ g α. 39

4 4 Sectio 3 Usig this i the r.h.s of Eq. 36 gives F g δf g = q [ δφg m X v ε Ω δφ g α ε + ] B µ f g 4 Followig Che s book, to make cotact with the low-frequecy limit, we would like to remove the /α terms i r.h.s of the above equatio. Thus we let δf g = q m δφ g ε + f g +δg g 41 B µ the substitute this ito Eq. 4 gives a equatio for δg g [ q F g m δφ g ε + ] f g + F g δg g = q [ δφg B µ m X v ε Ω δφ g α f g =, the above equatio is reduced to Usig F g ε + B µ ε + B µ B µ q f g F g m δφ g + F g δg g = q m [ δφg X v ε Ω δφ g α ε + ε + ] B µ f g. 42 ] B µ f g, 43 Usig the form of F g give by Eq. 27 i the above equatio gives ε + f g B µ t + v Ω q X α m δφ g + F g δg g = q [ δφg m X v ε Ω δφ g α ] f g, which ca be simplified to ε + [ q f g B µ m t + v F g δg g = q δφg m X v ε F g δg g = q [ fg m ε X f g δφ g t ] δφ g + F g δg g = q δφg m ] δφ g ε + [ t + v X + f g B µ t +v X X v ε which agrees with Eq. III.2.7 i Che s book[1]. Notig that δg g must be a periodic fuctio i the gyrophase agle α, i.e., thus, δg g ca be expressed as B µ f g, f g ε + δφ g ], 44 δg g X, µ, ε, α+,, t=δg g X, µ, ε, α,, t, 45 =+ δg g = δg g exp iα, 46 where δg g is idepedet of α. Similarly, δφ g X, ε, µ,, α, t must be a periodic fuctio i the gyrophase agle α, thus, δφ g ca also be expressed as =+ δφ g = δφ g exp iα, 47 where δφ g is idepedet of α. Substitutig the above expressios for δg g ad δφ g ito Eq. 44, yields the followig equatio for δg g F g δg g + iω t v X δφ g. + v X δg g = q m fg ε t + f g B µ t + 48

5 Solutio to the liearized equatio i the electrostatic limit 5 I obtaiig the above equatio, we have made use of the fact that differet harmoics are ot coupled. We ote i passig that 1 δg g = δg g. 49 Eq. 48 is similar to the umagetic case, hece ca be readily solved by Laplace trasformatio i time ad Fourier i space. Usig the followig otatio δâ g ω, k L p tf r X[δA g X, t] 5 Eq. 48 is solved to give δĝ g = 1 q f g ω k v Ω m ε ω + f g ω k v B µ δφˆg. 51 Now, i order to make cotact with later discussios o ouiform plasmas where v is ot a costat of the motio due to varyig B, we wat to remove the parallel propagator, /X, from the r.h.s of Eq. 44. Thus, we furhter write v X f g δg g = q m δφ g B dµ + δh g. 52 Substitutig this ito Eq. 44 yields a equatio for δh g + v t Ω q X α m δφ f g g + F g δh g = q B dµ m ] fg δφ g, B µ v X f g q m B dµ δφ g, t + v X f g q m B dµ F g δh g = q m Ω δφ g + F g δh g = α Ω α δφg t The, for the th harmoic i α, we obtai t + v + iω δh g X = q m δφ g + F g δh g = q m f g ε + Ω δφ g α fg ε q m fg δφg t fg δφ g ε t δφ g t, f g ε + [ t + + f g B µ t + ε f g, 53 B µ t iω f g B dµ δφ g, 54 which further gives Laplace i time ad Fourier i space δĥ g = q 1 fg m ω k v Ω ε ω + Ω f g δφˆg B dµ. 55 Usig Eqs. 41 ad 52, δf g ca be expressed i terms of δh g as δf g = q m δφ g ε f g + δh g, 56 which ca be further writte as δf g = q m δφ f g g ε + δh g exp iα. 57 Laplace trasformig i time ad Fourier trasformig i space, the above equatio is writte as δfˆg = q f g m δφˆg ε + δĥ g exp iα. 58

6 6 Sectio 3 Substitutig δĥ g give by Eq. 55 ito the obve equatio, gives δfˆg = q f g m δφˆg ε q m 1 fg ω k v Ω ε ω +Ω f g δφˆg exp iα, 59 B dµ For the electromagetic field, we have δa = δax, t. Trasformig to guidig-ceter coordiates, δa g δax, t=δa g X, V, t. 6 Note that δa g depeds o V. We ow derive the relatio betwee δâ ad δâ g, δa L 1 p ωf 1 dω d r kδâ = 3 k 3δÂω, kexp iωt + ik x = δa g = L 1 p ωf 1 r kδâ g dω d = 3 k 3δÂ gω, k, V exp iωt + ik X 61 Usig we obtai where X = x + v e Ω, 62 δâ g expil k =δâ, 63 L k =k v e Ω Without ay loss of geerality, we defie k = k e 1 + k e, the we have L k = λsiα, where λ = k /Ω, α is the gyrophase agle, which is defied as the icluded agle betwee ad e 1. Usig the idetity exp ± iλsiα= J λexp ± iα 65 i Eq. 63 gives 64 δâ g = δâexp il k = δâexp iλsiα = δâj λexp iα 66 For the electrical potetial, the above equatio is writte as δφˆg = δφˆj λexp iα 67 Comparig the above equatio with Eq. 47, we obtai δφˆg = δφˆj λ. 68 Usig this i Eq. 59 gives δfˆg = q f g m δφˆg ε q 1 fg m ω k v Ω ε ω + Ω f g δφˆj λexp iα, 69 B dµ Usig Eq. 67, the above equatio is writte as δfˆg = q δφˆf g J λexp iα q m ε m δφˆ 1 fg ω k v Ω ε ω + Ω f g J λexp iα, 7 B dµ

7 Solutio to the liearized equatio i the electrostatic limit 7 Usig that δfˆgexpiλsiα =δfˆ, Eq. 7 is writte as δfˆ = q δφˆf g m ε q m δφˆ Defie the gyrophase average J λexp iα +iλsiα 1 fg ω k v Ω ε ω +Ω f g J λexp iα + iλsiα 71 B dµ δfˆ the Eq. 71 ca be itegrated to give δfˆ = q δφˆf g J λ 1 O m ε q m δφˆ iλsiαdα O = 1 1 ω k v Ω dαδfˆg, 72 exp iα + iλsiαdα fg ε ω + Ω f g B dµ J λ 1 exp iα + We ote that the itegral represetatio of the Bessel fuctio J λ = 1 e iα λ siα dα, 74 the, Eq. 73 is writte as δfˆ = q δφˆf g J 2 O m ε λ q m δφˆ Usig δfˆ J 2 λ=1, the above equatio is writte [ = q O m δφˆ f g ε which agrees with Eq. III.2.23 i Che s book[1]. 3.1 Dispersio relatio Poisso s equatio is Laplace i time ad Fourier i space, we obtai k 2 δφˆ= 1 ε where the perturbed desity is give by ˆj = fˆjd 3 v = B dεdµ fˆj dα v B dεdµ v = Usig Eqs. 76 ad 79, Eq. 78 is writte as k 2 δφˆ= 2 [ 1 q j B dεdµ δφˆ ε m j v j 73 1 fg ω k v Ω ε ω +Ω f g J 2 B dµ λ. 75 J 2 λ fg ω k v Ω ε ω + Ω f ] g, 76 B dµ 2 δφ= 1 q j δ j. 77 ε j j f g ε q j δˆj, 78 δfˆj O 79 J 2 λ fg ω k v Ω ε ω + Ω f ] g, B dµ 8

8 8 Sectio 4 from which we obtai the dispersio relatio D e.s = 1+ j χ j = 81 where χ j, the j th-species susceptibility, is give by χ j = 1 2 q j 1 [ B dεdµ f g ε m j k 2 v ε = ω 2 pj 1 [ B dεdµ f g k 2 j v ε J 2 λ ω k v Ω J 2 λ ω k v Ω fg ε ω + Ω f ] g B dµ ], fg ε ω + Ω f g B dµ where ω 2 pj = j e 2 /m j ε, j is the equilibrium desity of the jth-species. Usig J 2 λ = 1, the above equatio ca also be writte as χ j = ω 2 pj 1 k 2 j = ω 2 pj 1 k 2 j Ω f g B dµ ] = ω 2 pj 1 k 2 j [ B dεdµ v [ B dεdµ v B dεdµ v J 2f g ε + which agrees with Eq. III.2.26 i Che s book[1]. J 2 λ fg ω k v Ω ω +k v + Ω ω k v Ω J 2 f g ε + ε ω +Ω f ] g B dµ J 2 λ ω k v Ω fg ε ω + J 2 [ λ k ω k v Ω v +Ω f g ε + Ω f ] g, 82 B dµ 4 Kietic theory of low-frequecy Magetohydrodyamic waves Accordig to Eqs. 41 ad 44, we have δf g = q m δφ g ε + ad t + v Ω X α δg g = q [ δφg f g m t ε + B µ f g + δg g, 83 t + v X ] fg δφ g. 84 B µ We cosider Ω/α term to be at the fastest times scale, at the order of O1, all other terms, amely, /t ad v /X, are at the order of Oη, where η is a small parameter. Expadig δg g as δg g =δg g + δg g1 + where δg g Oη. Similarly for δφ g, i.e., δφ g = δφ g + δφ g1 + Substitutig the expasios ito Eq. 84, we have, to the order of O1,, α δg g =, 87 which idicates δg g is idepedet of gyrophase α. To the ext order, Oη, we have t + v δg g Ω X α δg g1 = q [ ] δφg f g m t ε + t +v fg δφ g. 88 X B µ

9 Bibliography 9 Makig use of the periodicity of δg g1 i gyrophase α, we ca average the above equatio over gyrophase to elimiate the last term o the l.h.s, which gives otig that δg g is idepedet of α t +v δg g = r.h.s X O, 89 where r.h.s O 1 dαr.h.s = q 1 δφg dα m t = q m δφg t f g ε + [ f g ε + t + v X [ t + v X ] fg δφ g B µ δφ g ] fg B µ, 9 where, i obtaiig the last equality, we have used the fact that f g is idepedet of α. Also we have used that 1 δφ g dα = δφ g, 91 where δφ g is the expasio coefficiet whe expadig δφ g as series of α harmoics. Equatio 89 alog with Eq. 9 gives the electrostatic low-frequecy liear grokietic equatio for uiform magetized plasmas. The equatios agree with Eq.III.7.7 ad Eq. III.7.8 i Che s book[1]. Notig that δφˆg = J λφˆ 92 ad makig Laplace trasformatio i time ad Fourier i space to both sides of Eq. 89, we obtai iω + ik v δĝ g = q f [ iωj g λφˆ m ε + ] fg iω +k v J λφˆ 93 B µ δĝ g = q m J λφˆ ω ω k v f g ε + f g B µ 94 Bibliography [1] Liu che. Wave ad Istabilities i Plasmas. World Scietific Pub Co Ic, 1987.

Physics 232 Gauge invariance of the magnetic susceptibilty

Physics 232 Gauge invariance of the magnetic susceptibilty Physics 232 Gauge ivariace of the magetic susceptibilty Peter Youg (Dated: Jauary 16, 2006) I. INTRODUCTION We have see i class that the followig additioal terms appear i the Hamiltoia o addig a magetic