Magnetic Fields on 2D and 3D Spherestaff.norbert/JNMP-Conference-2013/Cabrerizo.pdfKilling magnetic elds on the 3D unit Sphere References Magnetic Fields on 2D and 3D Sphere Jose L

Preliminaries on magnetic eldsMagnetic elds on surfaces

Magnetic elds on 3D manifoldsKilling magnetic elds on the 3D unit Sphere

References

Magnetic Fields on 2D and 3D Sphere

Jose L. Cabrerizo

University of Seville

Nordfjordeid, June 8, 2013

Jose L. Cabrerizo Magnetic Fields on 2D and 3D Sphere



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Contents

1 Preliminaries on magnetic elds

2 Magnetic elds on surfaces

3 Magnetic elds on 3D manifolds

4 Killing magnetic elds on the 3D unit Sphere

5 References




References

Preliminaries

Our planet's magnetic eld




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Preliminaries

Maxwell's equations (1861)

∇ · E = ρε0

Gauss's law

∇ · B = 0 Gauss's law for magnetism (no monopoles)

∇∧ E = −∂B∂t Maxwell-Faraday

∇∧ B = µ0J + µ0ε0∂E∂t Ampère's law

E electric eld, B magnetic eld, ρ total charge, ε0 electricconstant, µ0 magnetic constant, J current density




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Preliminaries

Lorentz Force Law

F = q (E + v(t) ∧ B)E=0=⇒ Fm =

dv(t)

dt= q v(t) ∧ B

Fm=magnetic force, E=electric eld, q=charge, v=velocityvector, B=magnetic eld




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Preliminaries

Example in R3

Assume B = (0, 0, h), h ∈ R and v(t) = (v1(t), v2(t), v3(t))then dv(t)/dt = v(t) ∧ B =⇒ dv1(t)/dt = ω v2(t), dv2(t)/dt =−ω v1(t), dv3(t)/dt = 0 ⇒ magnetic curve:γ(t) =

(x01

+ r sin(ωt + α), x02

+ r cos(ωt + α), x03

+ v03t)




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Preliminaries

2-Form associated to B

Dene on OXY the 2-form F (X ,Y ) = 〈B ∧ X ,Y 〉

⇒ F = hdx1 ∧ dx2, dF = 0

Dene operator Φ : F (X ,Y ) = 〈Φ(X ),Y 〉

⇐⇒ Φ(X ) = B ∧ X

Lorentz force equation:dv(t)

dt= Φ(v(t))




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Preliminaries

A general setting

A magnetic eld on (Mn, g), is a closed 2-form F on Mn. TheLorentz force of F is the skew-symmetric operator Φ given

by F (X ,Y ) = g(Φ(X ),Y )

A curve γ(t) in (Mn, g) is a magnetic curve of (Mn, g ,F ) if

∇γ′γ′ = Φ(γ′) (Landau-Hall equation)

where ∇ is the Levi-Civita connection of g (compare withdv(t)

dt= Φ(v(t))




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Preliminaries

Consequences

F = 0 ⇐⇒ Φ = 0 ⇐⇒ ∇γ′γ′ = Φ(γ′) = 0 ⇐⇒ geodesics

If F 6= 0 existence and uniqueness theorem for geodesicsremains true for magnetics curves

Conservation law: particle evolve with constant speedalong a magnetic trajectory:

d

dtg(γ′, γ′) = ∇γ′g(γ′, γ′) = 2g(∇γ′γ′, γ′) = 2g(Φ(γ′), γ′) ≡ 0

Non-homgeneity: γ(t) magnetic curve of (M, g ,F ) thenγ(λt) magnetic curve of (M, g , λF ) and of (M, (1/λ)g ,F )(λ > 0)




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Preliminaries

Consequences




d






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Preliminaries

Consequences




d






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Preliminaries

Consequences




d






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Preliminaries

Consequences




d






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Preliminaries

Variational Principle

Geodesics ≡ critical points of an energy action ≡represent the trajectories for free fall particles movingunder the inuence of only gravity

Magnetic ows locally are also critical points of afunctional: ∃U ⊆ M such that F = dω in U, Γ = space ofcurves from p to q, then

L(γ) =1

2

∫γg(γ′, γ′)dt +

∫γω(γ′)dt

and the Lorentz equation is the Euler-Lagrange equationassociated to L




References

Preliminaries

Variational Principle

Geodesics ≡ critical points of an energy action ≡represent the trajectories for free fall particles movingunder the inuence of only gravity

Magnetic ows locally are also critical points of afunctional: ∃U ⊆ M such that F = dω in U, Γ = space ofcurves from p to q, then

L(γ) =1

2

∫γg(γ′, γ′)dt +

∫γω(γ′)dt

and the Lorentz equation is the Euler-Lagrange equationassociated to L




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Preliminaires

The Goal

For a given magnetic eld F in (Mn, g), nd all themagnetic curves on Mm

nD case: local chart in Mn and "translate"Landau-Hallequation to Rn, solve here equation and pull back to Mn

the curve

nD case: consider Mn → Rm (Nash Emb. Thm) and writethe Landau-Hall equation in Rm =⇒ Gauss-Weingartenformulas of submanifold theory, the tangentialcomponent to Mn of these equations can be intrinsicallystudied to get information of γ in Mn (S3 → R4)

2D case: by using only intrinsic geometry of the surface




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Preliminaires

The Goal



the curve






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Preliminaires

The Goal



the curve






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Preliminaires

The Goal



the curve






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Magnetic elds on surfaces

Frenet reference

(M2, g) Riemannian surface, J complex structure, areaelement Ω2 such that Ω2(X , JX ) = 1, X unit vector eld.γ regular curve, ‖γ′‖2 = g(γ′, γ′) = 1 constant, Frenetreference T = γ′,N = JT then

∇TT = κN, ∇TN = −κT (Frenet equations)

It is clear that any magnetic eld on M2 is F = f Ω2, f thestrength of F . We see that Φ(X ) = f (JX ),Φ(JX ) = −fX ,therefore the matrix of Φ with respect to the basis X , JX is

Φ ≡(

0 −ff 0

)Jose L. Cabrerizo Magnetic Fields on 2D and 3D Sphere



References


Frenet reference

In particular, along the magnetic curve γ with curvature κthe Lorentz force is

Φ(T ) = Φ(γ′) = ∇γ′γ′ = ∇TT = κN

Then f = κ =⇒ Φ ≡(

0 −κκ 0

)Therefore we have,

Theorem

The curvature of the normal magnetic curves is given byκ = f




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Uniform magnetic elds on surfaces

A magnetic eld F = f Ω2 with constant strengthf = µ ∈ R− 0, is called a uniform magnetic eld.Extensively considered in the literature from dierent pointsof view

Corollary

Let F = µΩ2 be a uniform magnetic eld, with constantstrength µ 6= 0 on (M2, g). The normal magnetic curves arethe curves of constant curvature κ = µ.




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Theorem 1

Flat surfaces (K ≡ 0): the magnetic curves of a uniformmagnetic eld F = µΩ2 are geodesic circles withgeodesic radius 1/|µ|




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Theorem 1

Flat surfaces (K ≡ 0): the magnetic curves of a uniformmagnetic eld F = µΩ2 are geodesic circles withgeodesic radius 1/|µ|




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Theorem 2

In the 2D-sphere S2(r) (K = 1/r2): magnetic curves aregeodesic circles with plane radius ρ = r/

√1 + r2µ2

(ρ < r) =⇒ no great circle can be a magnetic curve ofF = µΩ2 6= 0 on S2(r)




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Theorem 2

In the 2D-sphere S2(r) (K = 1/r2): magnetic curves aregeodesic circles with plane radius ρ = r/

√1 + r2µ2

(ρ < r) =⇒ no great circle can be a magnetic curve ofF = µΩ2 6= 0 on S2(r)




References


Theorem 3 (A. Comtet, Ann. Phys., 1987)

In the Poincaré half-plane H2(−c): (a) If |µ| >√c,

geodesic circles; (b) If |µ| <√c , non-closed curves

which intersect the boundary line ∂H2(−c); (c) when|µ| =

√c they are tangent to this boundary, and so they

are horocycles




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Theorem 3 (A. Comtet, Ann. Phys., 1987)

In the Poincaré half-plane H2(−c): (a) If |µ| >√c,

geodesic circles; (b) If |µ| <√c , non-closed curves

which intersect the boundary line ∂H2(−c); (c) when|µ| =

√c they are tangent to this boundary, and so they

are horocycles




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Magnetic elds on 3D manifolds

Key facts in 3D

Let (M3, g) be a 3D oriented manifold with volume form Ω3

2-forms and vector elds are in 1-1 correspondence:F =⇒ ω = ?F is a 1-form =⇒ denes a vector eld U byg(U,X ) = ω(X ). Conversely U ?ω = iUΩ3 = F

Magnetic elds come from divergence-free vector elds:LUΩ3 = d (iU Ω3) =dF= div(U)Ω3 =⇒ F is closed ⇐⇒div(U) = 0 (magnetic eld: U or F )




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Key facts in 3D







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Key facts in 3D







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Key facts in 3D

A cross product of vector elds on (M3, g) can bedened: X ,Y ∈ X(M3) dene X ∧ Y as follows

g(X ∧ Y ,Z ) = Ω3(X ,Y ,Z )

If X =∑

X iei , Y =∑

Y jej , Z =∑

Z kek the followingformulas are easily proved

X ∧ (Y ∧ Z ) = g(X ,Z )Y − g(X ,Y )Z ,

g(X ∧ Y ,X ∧ Z ) = g(X ,X ) g(Y ,Z )− g(X ,Y ) g(X ,Z )




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Theorem

The Lorentz force equation in (M3, g) is now written as∇γ′γ′ = U ∧ γ′

In fact, the Lorentz force Φ of F = iUΩ3 is obtained fromg(Φ(X ),Y ) = F (X ,Y ) = (iU Ω3)(X ,Y ) = Ω3(U,X ,Y ) =

= g(U ∧ X ,Y ) =⇒ Φ(X ) = U ∧ X =⇒ ∇γ′γ′ = U ∧ γ′




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Let us consider on (M3, g) a divergence-free vector eld U

and a magnetic curve γ(t). If T = γ′,N,B is the Frenetframe of γ, the Frenet equations are written as

∇TT = κN, ∇TN = −κT + τB, ∇TB = −τN

Easy computations give

Proposition

The Lorentz force Φ of the magnetic eld satises

Φ(T ) = κN, Φ(N) = −κT + ωB, Φ(B) = −ωN

where ω is a kind of slope of the magnetic curves withrespect to the magnetic eld




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Theorem

Let γ(t) be a magnetic curve of the magnetic eld U. ThenU(t) = ω(t)T (t) + κ(t)B(t) along γ

Proof. Let us write U(t) = a1(t)T (t) + a2(t)N(t) + a3(t)B(t),and assume U doesn't vanish on γ. But0 = U ∧ U = Φ(U) = (a1κ− a3ω)N − a2κT + a2ωB =⇒ a2 = 0,otherwise ω(t0) = κ(t0) = 0 for some t0 and so Φ = 0 atγ(t0) =⇒ U(t0) = 0. But from previous Prop. we have

0 = Φ(U) = a1Φ(T ) + a3Φ(B) = (a1κ− a3ω)N

which means a1 = ω and a3 = κ.




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Killing magnetic elds on 3D manifolds

Denitions

A Killing vector eld U on a Riemannian manifold (Mn, g), isa vector eld that generates local ows of isometries:LU g = 0 =⇒ 0 = (LU g)(X ,Y ) = g(∇XU,Y )+g(∇YU,X ) = 0,which is a useful characterization. Therefore we havediv(U)=

∑g(∇eiU, ei ) = 0 =⇒ a Killing vector eld is

divergence-free =⇒

when n = 3 a Killing vector eld denes a magnetic eld FUwhich will be called a Killing magnetic eld. In particular,uniform magnetic elds (∇V = 0) are obviously Killing. Fromnow on, we will refer to U or FU as the Killing magnetic eld




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Theorem (and new conservation law)

A magnetic eld U in a 3D manifold is Killing if and only ifalong any magnetic curve γ, g(U, γ′) is a constant

Proof. U Killing eld and γ magnetic curve =⇒g(∇γ′U, γ′) = 0 and ∇γ′γ′ = Φ(γ′) = U ∧ γ′ =⇒d

dtg(U, γ′) = g(∇γ′U, γ′)+g(U,∇γ′γ′) = 0 =⇒ g(U, γ′) = const.

Conversely: assume γ is a magnetic trajectory of a magneticeld U such that γ(0) = p, γ′(0) = v =⇒

0 =d

dtg(U, γ′) = g(∇γ′U, γ′) + g(U,U ∧ γ′) = g(∇γ′U, γ′).

Therefore, g(∇vU, v) = 0 =⇒ U is Killing.Jose L. Cabrerizo Magnetic Fields on 2D and 3D Sphere



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Killing magnetic elds also satisfy a certain symmetryproperty between its magnetic trajectories: let ϕt denote a(local) ow of a Killing vector eld U.

(a) If γ is any magnetic trajectory of U then ϕt γ isanother magnetic trajectory of U;

(b) For any magnetic trajectory γ of U the constantsassociated to the two conservation laws for γ and ϕt γare equal.

Theorem (Bianchi, 1918)

A Killing vector eld U on a Riemannian manifold (Mn, g)has constant length if and only if every integral curve of U isa geodesic in (Mn, g).




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Killing magnetic elds on the 3D unit Sphere

The 3D unit sphere

S3 = z = (x1, x2, x3, x4) ∈ R4 : |z | = x21 + x22 + x23 + x24 = 1

endowed with metric g induced from i : S3 −→ (R4, g0). Thequaternionic structure of R4 can be used to writez = x1 + ix2 + jx3 + kx4, with i2 = j2 = k2 = ijk = −1 =⇒

S3 = z = x1 + ix2 + jx3 + kx4 ∈ R4 : |z | = x21 + x22 + x23 + x24 = 1.

The following vectors are obtained by rotating z:iz = −x2 + ix1 − jx4 + kx3, zi = −x2 + ix1 + jx4 − kx3jz = −x3 + ix4 + jx1 − kx2, zj = −x3 − ix4 + jx1 + kx2kz = −x4 − ix3 + jx2 + kx1, zk = −x4 + ix3 − jx2 + kx1




References


Let us consider the six vector elds in R4 whose componentsw.r.t the standard frame e1, e2, e3, e4 are the same as thoseof iz , jz , kz , zi , zj , zk , that is,

U = (−x2, x1,−x4, x3), G = (−x3,−x4, x1, x2)V = (−x3, x4, x1,−x2), H = (−x2, x1, x4,−x3)W = (−x4,−x3, x2, x1), K = (−x4, x3,−x2, x1)

All of them are dierentiable unit vector elds, orthogonal toz = (x1, x2, x3, , x4) =⇒ unit vector elds on (S3, g). Besides,U,V ,W and G ,H,K are two orthonormal sets of vectorelds.




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Theorem

The unit vector elds U,V ,W ,G ,H,K are Killing on S3.

Proof. ∇, ∇ covariant derivatives on (R4, g0) and (S3, g),then ei , ej , i , j = 1, 2, 3, 4 direct computation gives

g0(∇eiU, ej) + g0(∇ejU, ei ) = 0,

g0 bilinear =⇒ result extends to any X =∑

X iei , Y =∑

Y jej=⇒ U Killing on R4.Now, Gauss and Weingarten formulas of submanifold theory:

∇XY = ∇XY + II (X ,Y )z , ∇X z = −AzX

II = second fundamental form, Az = shape operator on S3Jose L. Cabrerizo Magnetic Fields on 2D and 3D Sphere



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But II (X ,Y ) = g(∇XY , z) = −g(∇X z ,Y ) = g(AzX ,Y ) and S3is totally umbilical =⇒ AzX = −X . ThereforeII (X ,Y ) = −g(X ,Y ) =⇒ ∇XY = ∇XY − g(X ,Y )z . Inparticular, as U is Killing in (R4, g0), we get

0 = g0(∇XU,Y ) + g0(∇YU,X ) = g(∇XU,Y ) + g(∇YU,X )

=⇒ U Killing on S3

Corollary

The integral curves of U,V ,W ,G ,H,K in S3 are great circles.

It is easy to obtain a trajectory of U in S3 starting at t = 0

from z0 = (a, b, c , d) :α(t) = (acos(t)− bsin(t), asin(t) + bcos(t), ccos(t)− dsin(t),csin(t) + dcos(t)) (great circle)




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Remark

If X and Y are Killing then [X ,Y ] Killing =⇒ i(S3) Liealgebra with dimension at most (1/2)n(n + 1) = 6 (n = 3) =⇒B = U,V ,W ,G ,H,K is a basis of i(S3) and isomorphic toI (S3) group of isometries of the 3D sphere.

Take a vector eld in B, say U, and denote FU , ΦU theassociated Killing magnetic eld and Lorentz force,respectively. We have the following derivation formula.

Lemma

For any unit Killing vector eld in the basis B, say U, wehave ∇XU = ΦU(X ), ∀X ∈ X(S3).




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Proof. First, a direct computation gives

∇UU = −z , ∇VU = W , ∇WU = −VFrom G-W formulas

−z = ∇UU = ∇UU − g0(U,U)z =⇒ ∇UU = 0 (geodesics)

G-W equations with X = V ,Y = U give

∇VU = ∇VU−g(V ,U)z = ∇VU, ∇WU = ∇WU−g(W ,U)z = ∇WU

Finally, for any X = λU + µV + νW ∈ X(S3),

∇XU = λ∇UU + µ∇VU + ν∇WU = µW − νV .But ΦU(X ) = U ∧ X =⇒

ΦU(X ) = λU ∧ U + µU ∧ V + νU ∧W = µW − ν V ,and this proves the theorem




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Theorem

The magnetic curves γ of a Killing magnetic eld U in S3 arecurves of constant curvature κ0 and torsion τ0 = 1− ω0,therefore they are helices with axis the trajectories of U

Proof. U(t) = ω(t)T (t) + κ(t)B(t), g(U,T ) = ω(t) = ω0=⇒ 1 = g(U,U) = ω2

0+κ2 =⇒ κ2(t) = 1−ω2

0=⇒ κ(t) = κ0 =⇒

U(t) = ω0T (t) + κ0B(t)But ∇γ′U = ΦU(γ′) = U ∧ γ′ and Frenet formulas give

∇γ′U = ∇γ′(ω0γ′ + κ0B) = ω0κ0N + κ0(−τN) == κ0(ω0 − τ)N,

compare withU ∧ γ′ = (ω0T + κ0B) ∧ T = κ0N =⇒ ω0 − τ = 1⇐⇒ τ(t) is aconstant τ0 = ω0 − 1 which proves the theorem.




References


SuperNova 1987A: April 1, 2013: a team of astronomers led by the

International Centre for Radio Astronomy Research (ICRAR) have succeeded

in observing the death throws of a giant star in unprecedented detail:

"Supernova remnants are like natural particle accelerators, the radio emission

we observe comes from electrons spiralling along the magnetic eld lines and

emitting photons every time they turn", said Professor Lister Staveley-Smith,

Director of ICRAR




References

T. Adachi, Curvature bound and trajectories for magnetic elds on aHadamard surface, Tsukuba J. Math., 20 (1996) 225230.

M. Barros, J. L. Cabrerizo, M. Fernández, A. Romero, Magnetic vortexlament ows, J. Math. Phys., 48(2007) 08904

L. Bianchi, Lezioni sulla teoria dei gruppi continui niti di trasformazioni,Spoerri, Pisa (1918).

A. Comtet, On the Landau levels on the hyperbolic plane, Ann. Phys.,173(1987), 185209.

F. Harary, Graph Theory, Addison-Wesley, Reading, 1969.

D. A. Kalinin, Trajectories of charged particles in Kaehler magneticelds, Reports in Math. Phys., 39 No.3 (1997), 299-309.




References

A. Lopez-Almorox, Almost Kaehler structures on the tangent bundle ofa Riemannian manifold associated to the magnetic ow, Publ. R. Soc.Mat. Esp., 3 (2001) 133-150.

J. Nash, The embedding problem for Riemannian manifolds, Ann. ofMath. (2) 63, 20-63

S. Kobayashi, K. Nomizu Foundations of dierential geometry, Vol. I,Interscience Publishers, New York (1963).

S. Kobayashi, K. Nomizu Foundations of dierential geometry, Vol. II,Interscience Publishers, New York (1969).

M. P. Wojtkowski, Magnetic ows and Gaussian thermostats onmanifolds of negative curvature, Fund. Math., 163 (2000) 177191.

G. Zanardo, L. Staveley-Smith, C.-Y. Ng, B. M. Gaensler, T. M. Potter,R. N. Manchester, and A. K. Tzioumis, High-resolution radioobservations of the remnant OF SN 1987A at high frequencies,TheAstrophysical Journal, Vol. 767 No.2 (2013),doi:10.1088/0004-637X/767/2/98




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Magnetic Fields on 2D and 3D Spherestaff.norbert/JNMP-Conference-2013/Cabrerizo.pdfKilling magnetic elds on the 3D unit Sphere References Magnetic Fields on 2D and 3D Sphere Jose L