Notes on magnetic curves in 3D semi-Riemannian manifolds Zehra (Bozkurt) ÖZDEM · IR 1; · Ismail GÖK 2;Yusuf YAYLI 3and Faik Nejat EKMEKC · I 41 [email protected], 2 [email protected], 3 [email protected], 4 [email protected], Department of Mathematics, Faculty of Science, University of Ankara, TURKEY Abstract A magnetic …eld is de…ned by the property that its divergence is zero in three dimensional semi-Riemannian manifolds. Each magneti c …eld generates a magnetic ‡ow whose trajectories are curves called as magnetic curv es. In this paper, we inv estig ate the e¤ect of magnet ic …elds on the moving particle trajectories by variational approach to the magnetic ‡ow associated with the Killin g magnet ic …eld on three dimensio nal semi-Rieman nian manifolds. Then, we inv estig ate the trajectories of these magnetic …elds and give some characterizations and examples of these curves. 2000 Mathematics Subject Classi…cation. 37C10, 53A04. Key words and phrases. Special curves, vector…elds, fows, ordinary di¤erential equations 1 Introduction A ch arg ed par tic le moves alo ng a reg ular cur ve in 3-d ime nsi ona l spa ce. The tange nt , normal and binormal vector s describe kinematic and geomet ric propertie s of the particle. These vec tors a¤ect trajecto ry of the charg ed particle during motion of in a magnetic …eld. Also, time dimension a¤ects its trajecto ry . There fore, motion of the charge d parti cle in a magnetic vector …eld should b e investiga ted considering time dimension. In this article, we investigate e¤ects of magnetic …elds on charged particle trajectories by variational approach to magnetic ‡ow associated with Killing magnetic …eld on a three dimensional semi-Riemannian manifold M. A divergence free vector …eld de…nes a magnetic …eld in a three dimensional semi-Riemannian man- ifold M. It is kn own tha t V2 (Mn ) is Killing if and only ifL Vg = 0 or, equivalently, rV( p) is a skew-symmetric operator in T p(Mn ), at each point p 2Mn . It is clear that any Killing vector …eld on (Mn ; g)is divergence-free. In particular, ifn = 3, then every Killing vector …eld de…nes a magnetic …eld which will be called a Killing magnetic …eld(see for details ref.[3]). Lorentz force associated with the magnetic …eld Vis de…ned by (0 ) = V0 and trajectories called magnetic curves satisfy r 0 0 = (0 ) = V0 (1) 1
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8/9/2019 Notes on magnetic curves in 3D semi-Riemannian manifolds
where r is the Levi-Civita connection of the manifold M (in this article we call these curves as T-
magnetic curves to avoid confusion with other de…nitions) : Using Eq.(1) we can study the magnetic
…eld in a space which has non-zero sectional curvature C . So, this gives more important and realitic
approach than the classical approach. Also, this equality and the Hall e¤ect (explains the dynamics of
an electric current ‡ow in R3 when exposed to a perpendicular magnetic …eld V ) have some important
applications in analytical chemistry, biochemistry, atmospheric science, geochemistry, cyclotron, proton,
cancer therapy and velocity selector. Solutions of the Lorentz force equation are Kirchho¤ elastic rods.This provides an amazing connection between two apprently unrelated physical models and classical
elastic theory. The Lorentz force is always perpendicular to both the velocity of the particle and the
magnetic …eld created it. When a charged particle moves in a static magnetic …eld, it traces a helical
path and the axis of helix is parallel to the magnetic …eld. The speed of particle remains constant.
Since the magnetic force is always perpendicular to the motion, the magnetic …eld does no work on an
isolated charge. If the charged particle moves parallel to magnetic …eld, the Lorentzian force acting on
the particle is zero. When the two vectors (velocity and the magnetic …eld) are perpendicular to each
other, the Lorentz force is maximum (see for details ref. [2, 3, 4, 6, 7, 8, 9]).
When a charged particle moves along a curve in the magnetic …eld velocity (tangent vector), normal
and binormal vectors be exposed to the magnetic …eld. Then forces associated with the magnetic …eld
for motion in the normal and binormal directions of the curve are given by
(N ) = V N and (B) = V B;
and the trajectories of charged particle are changed according to this equation. For example; when a
charged particle moves in a static magnetic …eld in 3D Riemannian space the particle has a two di¤erent
paths. If one of the tangent or binormal vectors is exposed to this …eld, it traces a circular helix path.
On the other hand, if the normal vector is exposed the this …eld, it traces a slant helical path. Also,
their axes are parallel to the magnetic …eld (see ref. [5]).
2 Preliminaries
Let (M; g) be a 3dimensional semi-Riemannian manifold with the standard ‡at metric g de…ned by
g(X; Y ) = x1y1 + x2y2 x3y3 (2)
for all X = (x1; x2; x3); Y = (y1; y2; y3) 2 (M ):
The Lorentz force of a magnetic …eld F on M is de…ned to be a skew symetric operator given by
g((X ); Y ) = F (X; Y ) (3)
for all X; Y 2 (M ):
The T-magnetic trajectories of F are curves on M which satisfy the Lorentz equation
r 0 0 = ( 0): (4)
Furthermore, the cross product of two vector …elds X; Y 2 (M ) is given by
X Y = (x2y3 x3y2; x3y1 x1y3; x2y1 x1y2): (5)
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8/9/2019 Notes on magnetic curves in 3D semi-Riemannian manifolds
Proposition 2 Let be a unit speed spacelike or timelike space curve with (s)2 { (s)2 6= 0: Then
is a slant helix (which is de…ned by the property that the normal vector makes a constant angle with a
…xed straight line) if and only if { 2
("3{ 2 + "1 2)
{
0
is a constant function. Where "1 = g(T; T ); "2 = g(N; N ) and "3 = g(B; B) (see ref. [1]).
Proposition 3 Let be a unit speed non-null curve in semi-Riemannian manifold (M; g) with the Frenet apparatus fT; N; B; { ; g. Then the Serret-Frenet formula is given by
26664rT T
rT N
rT B
37775 =
26664
0 "2{ 0
"1{ 0 "3
0 "2 0
37775
26664
T
N
B
37775 (14)
where "1 = g(T; T ); "2 = g(N; N ) and "3 = g(B; B) [7].
3 Magnetic curves in 3D oriented semi-Riemannian manifolds
3.1 T-magnetic curves
In this section, we give some characterizations for T-magnetic curves in semi-Riemannian manifolds.
Proposition 4 Let be a unit speed non-null T-magnetic curve in semi-Riemannian manifold (M; g)
with the Frenet apparatus fT; N; B; { ; g. Then the Lorentz force in the Frenet frame written as
26664
(T )
(N )
(B)
37775
=
26664
0 "2{ 0
"1{ 0 "3$
0 "2$ 0
37775
26664
T
N
B
37775
(15)
where $ is a certain function de…ned by $ = g((N ); B).
Proof. Let be a unit speed T-magnetic curve in semi-Riemannian manifold (M; g) with the Frenet
apparatus fT; N; B; { ; g. From the de…nition of the magnetic curve we know that
(T ) = "2{ N:
Since (N ) 2 span fT ; N ; Bg ; we have
(N ) = T + N + B
Then using the following equalities
= "1g((N ); T ) = "1g((T ); N ) = "1{ ;
= "2g((N ); T ) = 0;
= "3g((N ); B) = "3$;
we get
(N ) = "1{ T + "3$B:
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8/9/2019 Notes on magnetic curves in 3D semi-Riemannian manifolds
Proposition 5 Let be a unit speed non-null curve in semi-Riemannian manifold (M; g). Then the
curve is a T-magnetic trajectory of a magnetic …eld V if and only if the vector …eld V can be written
along the curve as V = "3($T + { B): (16)
Proof. Let be a unit speed T-magnetic trajectory of a magnetic …eld V: Using Proposition 4 and
Eq.(7), we can easily see that
V = "3($T + { B)
Conversely, we assume that Eq.(16) holds. Then we get V T = (T ): So, the curve is a N-magnetic
trajectory of the magnetic vector …eld V .
Theorem 6 Let be a unit speed T-magnetic curve and V be a Killing vector …eld on a simply connected space form (M (C ); g): If the curve is one of the T-magnetic trajectories of (M (C ); g ; V ) then its
curvature and torsion hold the following equations
{ 2(
$
2 + ) + "1A = 0 (17)
"3{ 00 "2{ ($ + ) "3C { +
{ 3
2 B{ = 0 (18)
where C is curvature of the Riemannian space M and A; B are constant.
Proof. Let V be a magnetic …eld in a semi-Riemannian 3D manifold M . Then V satisfy Eq.(16).
Di¤erentiating Eq.(16) with respect to s, we have
rT V = "3$0T "1{ ($ + )N + "3{ 0B (19)
and di¤erentiation of Eq.(19) give us
r2T V = {
2($ + )T ("1${ 0 + 2"1{
0 + "1{ 0)N + ("3{ 00 "2{ $ "2{ 2)B: (20)
Lemma 1 implies that V (v) = 0: So, considering Eq.(19) we get
$ = const:
Then, if Eq.(19) and Eq.(20) are considered with V ({ ) = 0 in Lemma 1, we obtain
{ 2(
$
2 + ) + "1A = 0 (21)
Similarly, Eq.(19) and Eq.(20) are considered with V ( ) = 0 in Lemma 1, we can easily see that,
"3{ 00 "2{ ($ + ) "3C { +
{ 3
2 B{ = 0: (22)
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8/9/2019 Notes on magnetic curves in 3D semi-Riemannian manifolds
Proof. Let be a unit speed B-magnetic trajectory of a magnetic …eld V: Using the Proposition 20
and Eq.(7), we obtain Eq.35
V = "3 T + "3$2B
Conversely, we assume that Eq.(35) holds. Then we get (B) = V B. So, the curve is a B-magnetic
trajectory of the magnetic …eld V :
Theorem 22 (main result) Let be a unit speed B-magnetic curve and V be a Killing vector …eld on a simply connected space form (M (C ); g): If the curve is one of the B-magnetic trajectories of
(M (C ); g ; V ) then its curvature and torsion satisfy the equation
"3({ 0)2 + ("2 2 "2C
2 ){ 2 (2a"2 2 + 3"2C ){ +
{ 4
4 = : (36)
where C is curvature of the Riemannian space M , a; and are constants.
Proof. Let be a magnetic …eld in a 3D semi-Riemannian manifold. Then V satisfy Eq.(35). Di¤er-
entiating Eq.(35), we have
rT V = "3 T + "1 ({ $2)N + "3$02B (37)
Lemma 1 implies that V (v) = 0: So Eq.(37) gives us
0 = 0: (38)
If we di¤erentiate Eq.(37) with respect to s,
r2T V = { ($2 { )T + "1 ({ 0 2$0
2)N + ("3$002 + "2 2{ "2 2$2)B: (39)
Then, if Eq.(37) and Eq.(39) are considered with V ({ ) = 0 in Lemma 1, we obtain,
["1 ({ 0 2$02)]0 + g(R(V; T )T; N ) = 0
In particular, since C is constant, g(R(V; T )T; N ) = Cg(V; N ) = 0 we have
({ 0 2$02) = 0 (40)
Similarly, if we combine Eq.(37) and Eq.(39) wih V ( ) = 0 in Lemma 1 we get
[ 1
{ ("3$00
2 + "2 2{ "2$2 2 "2C$2)]0 +{ 2
2
0= 0; = const: (41)
If we integrate the Eq. (41) we obtain,
"3$002 + "2 2{ "2$2 2 "2C$2 +
{ 3
2 B{ = 0; = const: (42)
Finally, if Eq.(42) is combined with Eq.(40) and is multiplied by 2{ 0, we get