Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha
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1 Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v) Euler, Y- helicity, 2004 Yang-ills ), 0-22) Yang-ills. axwell : U) Y R 4 Â = A dx + A 2 dx 2 + A 3 dx 3 + φ dt F = d 4 Â = + Edt = dx 2 dx dx 3 dx + 3 dx dx 2 + E dx + E 2 dx 2 + E 3 dx 3 ) dt,. i = x A j k x A j, E k i = x φ i t A i. d 4 F = d 4 d 4 A = 0 3 i= x i = 0, R 3 d = 3 i= dx i x i x E j k x E k j + Ḃi = 0. d = 0, de + Ḃ = 0, ) div = 0, E + Ḃ = 0. 2)
2 2 j = j dx 2 dx 3 + j 2 dx 3 dx + j 3 dx dx 2 ρdx dx 2 dx 3 d 4 F = j dt + ρ R 3 R Hodge Hodge d E = ρdx dx 2 dx 3, d + Ė = j, 3) d E = ρ, d + Ė = j 4) div E = ρ, + Ė = j. 5) )2) 3)4) ρ j.2 Yang-ills 3 = R 4 vector potential )  = A + φdt = A dx + A 2 dx 2 + A 3 dx 3 + φdt, A k, φ sun).  Yang-ills d F   = 0, d  F  = 0, 6) Yang-ills dâ, d  4 covariant derivative Hodge dual, d A, d A covariant derivative Hodge dual F = + Edt, F A = ɛ ijk i dx j dx k, i = A k x j A j x k + [A j A k ], E = d A φ Ȧ = E idx i, E i = φ x i + [A i, φ] A i t, d  F  = 0 d A + [φ, E] + Ė = 0, d AE = 0. 7) dâfâ = 0 [ ianchi Identity ] d A E + [φ, ] Ḃ = 0, d A = 0. 8) 7), 8) Yang-ills 3 2
3 .3 Â = A + φ dt Ĝ = C R 4, sun)) ĝ Ĝ ĝ Â = ĝ Â ĝ + ĝ dĝ Fĝ Â = ĝ FÂ ĝ, dĝ Â ĝ ϕ ĝ) = ĝ dâϕ ĝ, Hodge Yang-ills 7), 8) ĝ Â = g Ag + g dg + g φ + ġg )g dt t g = gt) g A, φ) == g A, Ad g φ + ġg ) ). 2 3 Yang-ills 2. Yang-ills 3 φ = 0 E = Ȧ Yang-ills 7), 8) d A + Ė = 0, d AE = 0, 9) d A E Ḃ = 0, d A = 0. 0). d A + Ė = 0 d AE Ḃ = 0 vector potential ), Poisson. 2. d A E = 0 d A = 0 vector potential ) rduction moment map 0 ) moment map Lie G Lie G) Y- Y- 2.2 Y = 3 3 compact, G = SUn), n 2, sun) trace 0 n n. P G, A = A 3 3
4 A Ω, ad P ) = Ω, sun)) A A T A A = Ω, sun)). a, b T A A Ω, sun)) a, b) = T r a b, R = T A symplectic σ R = T A A, p ), p T A A, σ A,p) a, x), b, y) ) = b, x) a, y), ) a, x), b, y) T A,p) R = T A A T A A, R Φ a, x) ) a δφ A,p) = lim ΦA + ta, p + tx) ΦA, p)) x t 0 t δφ, δφ T δa δp AA Ω, adp ) ) ) a δφ 0 δφ A,p) = 0 δa, a, δφ A,p) x ) = ) δφ δp, x. δ HA, p) = 2, ) p, p), = F A,. 2) )a = lim δa t 0 F t A+ta F A ) = d A a, δp )x = x, δp ) a δh A,p) = d x A a, F A ) 2 + p, x) = a, d AF A ) + p, x) 3) 3), ) δh A,p) a x δh δa = d A, δh δp = p. ) = σ A,p) a, x), p, d AF A ) ). 4) X H p X H ) A,p) = d A ) = p A d A p. 5) Ȧ = p ṗ = d A 6) φ = 0 Y 9) p = E. 4
5 2.3 R, ω) G = Aut 0 P ) = Ω 0, AdP ) g A, p) = A + g d A g, g pg), g G 7) H LieG = Ω 0, adp ) ξ LieG ξ R ξ R A, p) = d dt t=0 exp tξ A, exp tξ E) = d A ξ, [ξ, p] ) T A,p) R. R J ξ dj ξ ) A,p) = σ A,p), ξ R ) J ξ A, p)) = d Ap, ξ) 0 8) ξ, η) 0 = T r ξ η 8) ) dj ξ a ) ) A,p) = lim d 0 t 0 t A+ta p, ξ) 0 d Ap, ξ) 0 = lim t 0 t p, d A+taξ d A ξ) = p, [a, ξ]) = a, [ξ, p]). ) dj ξ 0 ) A,E) = lim x t 0 t d Ap + tx), ξ) 0 d Ap, ξ) 0 ) = = d Ax, ξ) 0 = x, d A ξ). dj ξ ) A,p) a x ) = d A ξ, x) a, [ξ, p]) = σ A,p) a, x), d A ξ, [ξ, p]) ). δj ξ δa = d Aξ, δj ξ δp = [ξ, p] 8) J ξ J : R LieG) Lie G JA, p)) = { ξ J ξ A, p) = d A p, ξ ) 0 } p = E JA, p) = d A p 9) R 0 = {A, E) R; A A; JA, E) = 0} = {A, E) R; A A; d AE = 0} R = T A submanifold R 0, σ) G invariant coisotropic submanifold G R 0, σ) locally free G-orbit nullfoliation leaves R 0 /G, σ) reduced symplectic manifold arsden-weinstein reduction theorem). 5
6 Y- 9) d AE = 0 A LieG) current charge) {j, ρ} 2.2, 2.3 symplectic T A, σ) HA, p) =, ) E, E) 2 Hamiltonian Y d A Ė = 0, d A E = 0 A A A, = F A g A constant charge G reduced symplectic T A/G, σ) 3 Vorticity Clebsh parametrization. axwell 2), 4) A A A p = E Y- E Y- vorticity Y- E Y- G. 3. Y vorticity A T A π A. T A A, ) T A, A A, T A A,. A, ) T A T A,) T A) = T A A T AA Ω, adp ) Ω 2, adp ) 20) P = T T A) Poisson P A,, E) P, A A, E T A A, T A, Φ = ΦA, E, ) C P), δφ δe Ω, adp ) T A A, 2) a T A A Ω, adp ) ) a ΦA, E + ɛa, ) ΦA, E, ) dφ ) A,E,) = lim = 0 ɛ 0 ɛ a, δφ ) δe 22) δφ δ Ω2, adp ) T AA, 23) 6
7 β TA A Ω2, adp ) ) 0 ΦA, E, + ɛβ) ΦA, E, ) dφ ) A,E,) = lim = β ɛ 0 ɛ Poisson bracket {Φ, Ψ} vor = δφ δe, d A δψ ) δψ δ ) δe, d A δφ ) δ ) β, δφ ) δ 2. 24) 25) H = 2 E, E) +, ) 2 ) δh δe = E, δh δ = 26) Poissson Φ = {H, Φ} ) Ė Φ = dφ ) A,E,) = Ė, δφ ) + Ḃ, δφ ), Ḃ δe δ 2 {H, Φ} = E, d A δφ ) δ ) ) δφ δe, d A = d A E, δφ ) ) δφ δ 2 δe, d A Ė = d A, Ḃ = d A E, 27) Y 9), 0) 3.2 Clebsch parametrization ψ : T A A, p) E = p, = F A ) P = T T A) 28) Φ, Ψ C P) {Ψ ψ, Φ ψ} vor = {Ψ, Φ} R ψ 29) symplectic R, ω) Poisson P, {, } vor ) Poisson map } { Ȧ = E clebsch Ė = d A, Ė = d A = 30) Ḃ = d A E 7
8 3.3 Poisson P G = Ω 0, AdP ) infinitsimal T A θ, A, ) T A ) ) a a T A,) T A) = T A A TAA θ β A,) = T r a. 3) β β θ A,) a, a T A A ξ LieG A infiniteimal LieG A ξ, A) ξ A = d A ξ T A A A A d A ξ T A cotangent lift ) T A A, ) θ A,) d A ξ = T r d A ξ 32) Hamiltonian T A Hamiltonian vector field Hamiltonian vector field Φ XΦ) = { Φ, θ A,) d A ξ } P X ξ Lie G P infiniteimal momrnt map J J : P A, E, ) ξ θ A,) d A ξ ) LieG). 33) θ A,) d A ξ = T r d A ξ = T rd A ξ J = d A. 34) Y 0) d A = 0 A, E, ) P moment map J moment map = Lie G) 3.2, 3.3 Poisson vortex ) Clebch parametrization 8
9 4 incompressible flow Euler incompressible flow Euler ) volume preserving diffeomorphism divergence free ) 2) 3) ), 2) axwell Y- Clebsch parameter Helicity A-H. -W 4. G = SV ect) Euler vortex 4.. G = SV ect) R 3 SDiff) SV ect) = {v V ect); div v = 0, v // }. // v G = SV ect) v, u bracket [v, u] = v )u u )v δf v) G δv DF v)δv = lim ɛ 0 F v + ɛδv) F v) ɛ = δf δv v)δvdx3 F, G C G), v G, [ ] δf δg {F, G}v) = v v), δv δv v) dx 3. 35) G, {, }) Poisson Hv) = v vdx 3 2 Hamilton d F vt)) = {H, F }v) dt [ ] δf δf δv v) v dx3 = v δv v), v dx 3. SV ect) v v + v )v + p = 0, p, 36) 9
10 4..2 G G: vorticity vector fields G G {F, G} ν) = ν, [ δf δν, δg δν Poisson δf δν G DF ν)δν = lim ɛ 0 F ν + ɛδν) F ν) ɛ ]), ν G 37) = δν, δf δν Poisson P oment J : P G Poisson map= Clebsch parametrization) G = SV ect) 3 G G G Ω )/dω 0 ) G Z 2, ) vorticity vector fields ) G = {ω = v; v G}. u G v = u mod. f) v G. iot-savart s : vy) = Su) = ux) x y) d 3 x, 4π x y 3 : G S G. G ω, v) = G = SV ect) G G δh δω ω v d 3 x G G. 38) Hω) = 2 ω, Sω) ) = Sω) d δf F ω) = {H, F }ω) = ω, [v, dt δω ] ), v = Sω. v = Sω Lie L v [v, δf ] = L δω v δf δω {H, F }ω) = ω L v δf δω d3 x = 0 L v ω δf δω d3 x, F.
11 ω = L v ω, ω = v. 39) 36) 36) v = v ω + q ω = v = v ω) + q = ω )v v ) ω div v)ω + div ω)v = ω )v v )ω = L v ω. 39) Euler vortex vorticity ) 39) Euler G coadjoint orbit Lord Kelvin s circulation theorem: velocity vector field v vorticity vector field ω v = v volume preserving diffeomorphism diffeomorphism v velocity vector field vorticity Euler flow diffeomorphism velocity vector field vorticity 4.2 G G = SV ect) ) vorticity vector fields G G = Ω )/dω 0 ) 40) : [ν], v ) = Arnold inertia operator Av, w ) = νv) d 3 x, ν Ω ), v V ect). 4) A : G v ν = Av G 42) vx) wx) d 3 x, w, v V ect). 43), G G v = j v j x j ν = j v j dx j. inertia operator A G Euler 36) G ν = L A ν ν df )f Ω 0 ). 44) [A-K. IV.D] L v v Lie 44) G = Ω )/dω 0 ) Hν) = ν, 2 A ν ) δh = δν A ν, d dt F νt)) = {H, F }νt)) = ν, [A ν, δf δν ] ) = ν, ad δf A ν δν ),
12 ν = ad A ν ν. 45) SDiff) V ect) ), SDiff) ġ V ect). Ad g g Ad g v = g v, 46) Ad g ν = g ν, ad vν = L v ν, mod dω 0 ) 47) 45) 44) 4.3 R 3 ν = L A ν ν mod dω 0 ). 48) Ω )/dω 0 ) d Z 2, ) = {β Ω 2 ); dβ = 0, β = 0} G = SV ect) A Ω )/dω 0 ) G S d d G 49) G = SV ect) i vol. Z 2, ) Green G, ν Z 2, ) β = ν β = Gν Z 2, ) Svect) v v = Av Ω )/dω 0 ) SV ect) v = ω dv = ω = i ω vol Z 2, ), SV ect) v i v vol Z 2, ), Z 2, ) β, v)) = d Gβ, v) d 3 x, 2
13 G G Z 2, ) Poisson δf δβ G {F, G} = β, [ δf δβ, δg ] )), 50) δβ DF β)δβ = δβ, δf δβ Hβ) = 2 β d Gβ )) 5) v = δh DHβ)γ = γ δβ d Gβ DHβ)γ = γ, v)), γ Z 2, ), v = d Gβ, mod dω 0 ). ω = dv = β. v = δh β vorticity form velocity field δβ F = {F, H} = β, [ δf δβ, v ] )) = β, L δf v δβ )) = L vβ, δf δβ )) = DF β)l vβ. Euler vorticity vorticity ) ω = L v ω, dv = ω. 52) 39) 44) d vortex. 4.4 Clebsch parametrization F = C R 3 ), R c = compact support Radon measure. Z = F R c symplectic σλ, µ ), λ 2, µ 2 )) = λ µ 2 λ 2 µ, symplctic vector space F R c Functional H CZ) Frechet δh R δλ c D Hλ, µ)ξ lim ɛ 0 Hλ + ɛξ, µ) F λ, µ) ɛ = ξ δh R δλ 3, δh F δµ X H = ) δh, δh. δµ δλ 3
14 SDiff) Z moment map g λ, µ)x) = λg x), µx) ) J : Z G = Ω )/dω 0 ) J : λ, µ) λ dµ Ω )/dω 0 ). 53). v G = SV ect) F infinitesimal action Lie L v λ moment map < v, Jλ, µ) >= L v λ)x)µx)d volx) = λx)l v µ)x)d volx). L v µ = dµ)v < v, Jλ, µ) >= λx)dµv)x) d volx). Jλ, µ) = λdµ mod dω 0 ). G Z 2, ) Jλ, µ) = dλ dµ. F R c, σ) λ, µ) Z 2, ), {, }) symplectic F R c, σ) Poisson G, {, }) Poisson map λ, µ) Clebsch parametrization Clebsch parametrization Z = F R c, σ) symplectic Sp2, R) [-W] p.33) 4.5 Helicity v ω = v vorticity ) ω = v Helicity Hω) = v ω) d 3 x Hlicity vorticity ω ω v+ f V ect div, ) f ω d 3 x = 0. Hω) inertia operator A G v = Av G, ω = dv Hω) = v dv = v ω 54) 4
15 axwell Helicity Helicity), 54) H) = A da = A, = F A = da da + dφ) = da, d = 0 H) = da A A U) Chern-Simons Yang-ills 9) 0) Helicity Chern-Simons H) = T r AdA + 3 ) A3 = T r A 23 ) A3 55) G A/G vorticity Helicity HE) = T re d A E) 56) E = p g G p g pg 2,3 ) HE) orbit space A/G vorticity [ ] SDiff) axwell Ω ) semi direct product SDiff) Y Jackiw [-W]. arsden-weinstein: Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica 7D983), [A-K]. Arnold-Khesin: Topological methods in hydrodynamics, App. ath. Ser.25, Springer. [J], R. Jackiw: Lectures in fluid dynamics, A particle theorists view of supersymmetric, non-abelian, non-commutative fluid mechanics and d-branes. CR Ser in ath. Phys. Springer 5
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