notes on magnetic curves in 3d semi-riemannian manifolds

8/9/2019 Notes on magnetic curves in 3D semi-Riemannian manifolds

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Notes on magnetic curves in 3D semi-Riemannian manifolds

Zehra (Bozkurt) ÖZDEM·IR1;·Ismail GÖK2;Yusuf YAYLI3 and Faik Nejat EKMEKC·I4

[email protected], [email protected], [email protected],[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 magnetic …eld generates a magnetic ‡ow whose trajectories are

curves called as magnetic curves. In this paper, we investigate the e¤ect of magnetic …elds on

the moving particle trajectories by variational approach to the magnetic ‡ow associated with the

Killing magnetic …eld on three dimensional semi-Riemannian manifolds. Then, we investigate 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 charged particle moves along a regular curve in 3-dimensional space. The tangent, normal and

binormal vectors describe kinematic and geometric properties of the particle. These vectors a¤ect

trajectory of the charged particle during motion of in a magnetic …eld. Also, time dimension a¤ects its

trajectory. Therefore, motion of the charged particle in a magnetic vector …eld should be investigated

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 known that V 2 (M n) is Killing if and only if LV g = 0 or, equivalently, rV ( p) is a

skew-symmetric operator in T p(M n), at each point p 2

M n. It is clear that any Killing vector …eld on

(M n; g) is divergence-free. In particular, if n = 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 V is de…ned by

( 0) = V 0

and trajectories called magnetic curves satisfy

r 0 0 = ( 0) = V 0 (1)

1



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)

2



Then, the mixed product of the vector …elds X; Y; Z 2 (M ) is de…ned by

g(X Y; Z ) = dvg(X; Y; Z ) (6)

where dvg denotes a volume on M :

Let V be a Killing vector …eld and F V = {V dvg be the corresponding Killing magnetic force on M ,

where { denotes the inner product. Then the Lorentz force of the F V given as

(X ) = V X: (7)

for all X 2 (M ): Consequently, the T-magnetic trajectories determined by V are solutions of the

Lorentz force equation written as

r 0 0 = V 0: (8)

A unit speed curve is a T-magnetic trajectory of the magnetic …eld V if and only if V can be written

along as

V (s) = f (s)T (s) + g(s)B(s) (9)

where T and B are the tangent and binormal vectors of the curve , respectively (see ref. [4]).

Lemma 1 Let : I R ! M be a non-null immersed curve in a 3D semi-Riemannian manifold

(M; g) with sectional curvature C and V be a vector …eld along the curve . : ("; ") I ! M satisfy

(s; 0) = (s); @ @s (s; t) = V (s): In this setting, we have the following functions,

1. The speed function v(s; t) =@ @s (s; t)

;

2. The curvature function { (s; t) of t(s);

3. The torsion function (s; t) of t(s):

The variations of these functions at t = 0;

V (v) =

@v

@s(s; t)

t=0

= "1g(rT V; T ); (10)

V ({ ) =

@ {

@s (s; t)

t=0

= 2"2g(r2T V;rT T ) + 4"1{

2g(rT V; T ) (11)

+ 2"2g(R(V; T )T;rT T );

V ( ) =

@

@s(s; t)

t=0

= 2"2g((1={ )r3T V ({ 0={ 2)r2

T V (12)

+ "1("2{ + (C={ ))rT V "1C ({ 0={ 2)V ; B);

where "1 = g(T; T ); "2 = g(N; N ), "3 = g(B; B), R and C are curvature tensor and sectional curvature of M; respectively [4, 7] :

Also, if V (s) is the restriction to (s) of a Killing vector …eld then the vector …eld V satisfy following

condition

V (v) = V ({ ) = V ( ) = 0 (13)

[4].

3



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:

4



Similarly, we can easily obtain that

(B) = "2$N:

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)

5



Corollary 7 Let be a unit speed T-magnetic curve and V be a Killing vector …eld on a simply con-

nected space form (M (C ); g): If the function $ is equal to zero, then the curvature and torsion of the

curve is given by

"1{ 2 = A

2"3{ 00 + {

3 2"2{ 2 2"3C { 2B{ = 0:

So, we know that these two equations are just the Euler Lagrange equations for elasticae (see for details

ref. [7]) in M (C ):

It is easily seen that a T-magnetic curve is a general helix in Lorentzian 3-space. Some characteri-

zations and examples about it are detailed in the articles [4, 6].

3.2 N-magnetic curves

In this section, we de…ne a new kind of magnetic curve called N-magnetic curve in 3D semi-Riemannian

Manifold M . Moreover, we obtain some characterizations and examples of this curve. Also we draw the

image of these curves using the programme Mathematica.

De…nition 8 Let : I R !M be a non-null curve in semi-Riemannian manifold (M; g) and F V be

a magnetic …eld on M: We call the curve to be a N-magnetic curve if its normal vector …eld satisfy

the Lorentz force equation, that is,

r0N = (N ) = V N:

Proposition 9 Let be a unit speed non-null N-magnetic curve in semi-Riemannian manifold (M; g)

with the Frenet apparatus fT; N; B; { ; g. Then the Lorentz force in the Frenet frame can be written

as 26664

(T )

(N )

(B)

37775 =

26664

0 "2{ "3$1

"1{ 0 "3

"1$1 "2 0

37775

26664

T

N

B

37775 (23)

where $1 is a certain function de…ned by $1 = g((T ); B).

Proof. Let be a unit speed N-magnetic curve in semi-Riemannian manifold (M; g) with the Frenet

apparatus fT; N; B; { ; g. Since (T ) 2 span fT ; N ; Bg ; we have

(T ) = T + N + B:


= "1g((T ); T ) = 0;

= "2g((T ); N ) = "2g((N ); T ) = "2g(rT N; T ) = "2{ ;

= "3g((T ); B) = "3$1;

we get

(T ) = "2{ N + "3$1B:

6




(N ) = "1{ T "3 B;

(B) = "1$1T + "2 N:

Proposition 10 Let be a unit speed non-null curve in semi-Riemannian manifold (M; g). Then the

curve is a N-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 "3$1N + "3{ B: (24)

Proof. Let be a unit speed N-magnetic trajectory of a magnetic …eld V: Using the Proposition 9 and

Eq.(7), we can easily obtain Eq.24.

Conversely, we assume that Eq.(24) holds. Then we get V N = (N ): So, the curve is a

N-magnetic trajectory of the magnetic vector …eld V .

Theorem 11 (Main result) Let be a unit speed N-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 N-magnetic trajectories of

(M (C ); g ; V ) then its curvature and torsion satisfy the equation

"3{ 00 + "1

02

{ + 2"1 (

0

{ )0 "2C { +

{ 3 + "2{ 2

2 = (25)

where C is curvature of the Riemannian space M and is a constant.

Proof. Let V be a magnetic …eld in a semi-Riemannian 3D manifold M . Then V satisfy Eq.(24).

Di¤erentiating Eq.(24) with respect to s, we have

rT V = ("3 0 "2$1{ )T "3$01N + ("3{

0 + $1)B. (26)

Eq.(14) and di¤erentiation of Eq.(26) give us

r2T V = "2$0

1{ T + ("3$001 "1 { 0 + "2 2$1)N + ("3{

00 + 0$1 + 2$01 )B: (27)

Lemma 1 implies that V (v) = 0: So, considering Eq.(26) we get

$1 = "1 0

{ : (28)

Then, if Eq.(26) and Eq.(27) are considered with V ({ ) = 0 in Lemma 1, we obtain

"3$001 "1 { 0 + "2 2$1 + "1g(R(V; T )T; N ) = 0:

In particular, since C is constant, g(R(V; T )T; N ) = Cg(V; N ) = "3C$1 we have

"3$001 "1 { 0 + "2 2$1 + "2C$1 = 0: (29)

Similarly, Eq.(26) and Eq.(27) are considered with V ( ) = 0 in Lemma 1, we can easily see that,

[(1={ )("3{ 00 + 0$1 + 2$0

1 "2C { )]0 +{ 2 + "2 2

2

0= 0: (30)

7



Finally, integration of Eq.(30) and Eq.(28) give us

"3{ 00 + "1

02

{ + 2"1 (

0

{ )0 "2C { +

{ 3 + "2{ 2

2 =

where is a arbitrary constant.

Example 12 Let M be a three dimensional sphere and the curve be timelike N-magnetic curve. and

the constant function : If we get ,{ (s) = 1; = 3=2 and (s) = y(s) in Eq.(25) we obtain following

second-order nonlinear di¤erential equation

2y(s)02 + 4y(s)y00(s) y(s)2 = 0: (31)

Using the programme Mathematica we obtain solution of Eq.(31)

y(s) = c2e

2

p 23 log

0@e

p 6c1+e

p 32s

1As

p 6

:

Thus, the curve having the curvature { (s) = 1 and torsion (s) = c2e

2

p 23 log

0@e

p 6c1+e

p 32s

1As

p 6

: are N-

magnetic curves in three dimensional sphere.

Corollary 13 Let V be a Killing vector …eld on 3D semi-Riemannian manifold (M; g). Then each

trajectory of the magnetic …eld V makes an angle (s)with the normal vector …eld of the magnetic curve.

For the angle (s) we have the following cases;

Case 1. If N and V are spacelike vectors and these vectors span spacelike plane,

g(N; V ) = kN k kV k cos ref.[11].

Then using Eq. (24) and g(N; N ) = "2 = 1 we obtain

cos = g(N;V )kN kkV k = g(N;"3T "3$1N +"3{B)

kV k = "3$1q j"1 2+$2

1+"3{2j .

Case 2. If N and V are spacelike vectors and these vectors span timelike plane,

g(N; V ) = kN k kV k cosh ref.[11].Then using Eq. (24) and g(N; N ) = "2 = 1 we obtain

cosh = g(N;V )kN kkV k = g(N;"3T "3$1N +"3{B)

kV k = "3$1q j"1 2+$2

1+"3{2j .

Case 3. If N and V are timelike vectors in the same timecone,

g(N; V ) = kN k kV k cosh ref.[11].

Then using Eq. (24) and g(N; N ) = "2 = 1 we obtain

cosh = g(N;V )kN kkV k = g(N;"3T "3$1N +"3{B)

kV k = "3$1q j"1 2$2

1+"3{2j .

Case 4. If N spacelike (resp. timelike) and V timelike (resp. spacelike) vectors in the future timecone,

g(N; V ) =

kN

k kV

ksinh ref.[11].

Then using Eq. (24) and g(N; N ) = "2 we obtain sinh = g(N;V )kN kkV k = g(N;"3T "3$1N +"3{B)

"2kV k =

"3$1q j"1 2+"2$2

1+"3{2j .

Corollary 14 Let be a unit speed non-null N-magnetic curve in Lorentzian 3-space. If the function

$1 is non-zero constant, then the curve is a slant helix whose axis is the vector …eld :

Proof. We assume that is a N-magnetic curve in Lorentzian 3-space with non-zero constant function

$1, then from Eq.(28) and Eq.(29), we get

"1 0

{ = "3

{ 0

= $1 (32)

8



which implies that

"1{ 2 + "3 2 = const:

Also, Eq. (32) carry out the following equation with the di¤erent point of view, we get

{ 0 + { 0 = "3$1("3{ 2 + "1 2) = const:

or

{

2

("3{ 2 + "1 2)3=2

{

0 = const:

where $1 is a constant function. By Proposition 2 we obtain that is a slant helix in Lorentzian

3-space.

Example 15 We consider a spacelike N-magnetic curve with timelike normal vector in Lorentzian

space de…ned by

(s) =

0BBB@

p 32

2p 3(p 3+2)

cos(p

3 + 2)s +p 3+2

2p 3(p 32)

cos(p

3 2)s;p 32

2p 3(p 3+2)

sin(p

3 + 2)s +p 3+2

2p 3(p 32)

sin(p

3 2)s;

cos2s2p 3

1CCCA

The picture of the spacelike N-magnetic curve is rendered in Figure 1. The curve has the following

curvature and torsion

{ (s) = cos 2s;

(s) = sin 2s;

[10]. By using the Corollary 14 we can easily see that $1 = 2: So, is a N-magnetic curve.

Figure 1. Spacelike N-magnetic cuve with $1 = 2:

Example 16 We consider a spacelike N-magnetic curve with spacelike normal vector in Lorentzian

3space de…ned by

(s) = (4sinh3s

15 ;

1

40 sinh 8s +

2

5 sinh(2s);

1

40 cosh 8s 2

5 cosh(2s))

The picture of the spacelike N-magnetic curve is rendered in Figure 2. The curve has the following

curvature and torsion

{ (s) = 4cosh 3s;

(s) = 4sinh 3s;

9



[10]. By using the Corollary 14 we can easily see that $1 = 3: So is a N-magnetic curve.

Figure 2. Spacelike N-magnetic curve with $1 = 3:

Example 17 We consider a timelike N-magnetic curve in Lorentzian 3space de…ned by

(s) = (

p 3

2

cosh2s; 1

6

cosh 3s + 3

2

cosh(

s);

1

6

sinh 3s + 3

2

sinh(

s))

The picture of the timelike N-magnetic curve is rendered in Figure 3. The curve has the following

curvature and torsion given by

{ (s) =p

3sinh2s;

(s) =p

3cosh2s;

[10]. By using the Corollary 14 we can easily see that $1 = 2: So, is a N-magnetic curve.

Figure 3. Timelike N-magnetic curve with $1 = 2:

Corollary 18 Let be a non-null N-magnetic curve in Lorentzian 3space. If the function $1 is zero

then is a circular helix whose axis of the circular helix is the vector …eld V :

Proof. It is obvious from Eq. (23).

10



3.3 B-magnetic Curves

In this section, we de…ne a new kind of magnetic curve called B-magnetic curve in 3D semi-Riemannian

manifold: We obtain some characterizations and examples of this curve. Also, we draw the image of

these curves using the programme Mathematica.

De…nition 19 Let : I R ! M be a non-null curve in 3D semi-Riemannian manifold (M; g) and

F V be a magnetic …eld on M: We call the curve is a B-magnetic curve if the binormal vector …eld of

the curve satisfy the Lorentz force equation, that is,

r0B = (B) = V B (33)

Proposition 20 Let be a unit speed non-null B-magnetic curve in 3D semi-Riemannian manifold

(M; g) with the Serret-Frenet apparatus fT; N; B; { ; g. Then the Lorentz force in the Frenet frame

written as 2

6664(T )

(N )

(B)

3

7775 =

2

66640 "2$2 0

"1$2 0

"3

0 "2 0

3

7775

2

6664T

N

B

3

7775 (34)

where $2 is a certain function de…ned by $2 = g((T ); N ).

Proof. Let be a unit speed non-null B-magnetic curve in 3D semi-Riemannian space with the Frenet

apparatus fT; N; B; { ; g. Since we have,

(T ) = T + N + B:


= "1g((T ); T ) = 0;

= "2g((T ); N ) = "2$2;

= "3g((T ); B) = "3g((B); T ) = "3g(rT B; T ) = "3g("3 N ; T ) = 0:

we get

(T ) = "2$2N:


(T ) =

"1$2T

"3 B;

(T ) = "2 N:

Proposition 21 Let be a unit speed non-null curve in semi-Riemannian manifold (M; g): Then the

curve is B-magnetic trajectory of a magnetic …eld V if and only if the magnetic vector …eld V can be

written along the curve as

V = "3 T + "3$2B (35)

11



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

2"3{ 00{ 0 + (2"3 2 "2C 2B){{ 0 (2a"2 2 + 3"2C ){ 0 + {

3{ 0 = 0; = const: (43)

whose integration given

"3({ 0)2 + ("2 2 "2C

2 ){ 2 (2a"2 2 + 3"2C ){ +

{ 4

4 = ; = const: (44)

12



Example 23 Let M be three dimensional sphere and be a non-null B-magnetic curve. If we get

= a = 1; = 2 and = 5 in Eq.(44) then we obtain following …rst-order nonlinear ordinary

di¤erential equation

y0(t)2 + 4y(t)2 5y(t) + y (t)4

4 5 = 0

we obtain Figure 4 and Figure 5 by using the programme Mathematica:

Figure 4. Plots of sample individual solution

Figure 5. Sample solution family

Corollary 24 Let V be a Killing vector …eld on 3D semi-Riemannian manifold (M; g). Then each

trajectory of the magnetic …eld V makes an angle (s)with the binormal vector …eld of the magnetic

curve. For the angle (s) we have the following cases;

Case 1. If B and V are spacelike vectors and these vectors span spacelike plane,

g(B; V ) = kBk kV k cos ref.[11].

Then using Eq. (35) and g(B; B) = "3 = 1 we obtain

cos = g(B;V )kBkkV k = g(B;T +$2B)

kV k = $2q j"1 2+$2

2j:

Case 2. If B and V are spacelike vectors and these vectors span timelike plane,

g(B; V ) = kBk kV k cosh ref.[11].


cosh = g(B;V )kBkkV k = g(B;T +$2B)

kV k = $2q

j"1 2+$2

2

j

:

Case 3. If B and V are timelike vectors in the same timecone,g(B; V ) = kBk kV k cosh ref.[11].


cosh = g(B;V )kBkkV k = g(B;T +$2B)

kV k = $2q j 2$2

2j:

Case 4. If N spacelike (resp. timelike) and V timelike (resp. spacelike) vectors in the future timecone,

g(B; V ) = kBk kV k sinh ref.[11].

Then using Eq. (35) and g(B; B) = "3 we obtain

sinh = g(B;V )kBkkV k = g(B;"3T +"3$2B)

"3kV k = $2q j"1 2+"3$2

2j:

13



Corollary 25 Let be a non-null B-magnetic curve in a Lorentzian 3space with $2 constant, then

is a circular helix whose axis is the vector …eld V:

Proof. It is obvious from Eq. (36).

Example 26 We consider a timelike B-magnetic curve in Lorentzian 3space de…ned by

(s) = (cosh sp

2;

sp 2

; sinh sp

2)

[10]. The picture of the B-magnetic curve is rendered in Figure 6.

Figure 6. Timelike B-magnetic curve.

The curve has the following curvature and torsion

{ (s) = (s) = 1

2:

Using Corollary 25 we can easily see that is a B-magnetic curve.

Example 27 We consider a timelike B-magnetic curve in Lorentzian 3space de…ned by

(s) = ( 1p 2 sinh s;

sp 2 ;

1p 2 cosh s)

[10]. The picture of B-magnetic curve is rendered in Figure 7.

Figure 7. Spacelike B-magnetic curve.

The curve has the following curvature and torsion

{ (s) = (s) = 1p

2:

Using the Corollary 25 we can easily see that is a B-magnetic curve.

14



References

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[5] Bozkurt Z, Gök ·I, Yayl¬ Y, Ekmekci FN. A new Approach for Magnetic Curves in Riemannian

3DManifolds. J Math Phys 2014; 55: 1-12.

[6] Drut-Romaniuc SL, Munteanu MI. Killing magnetic curves in a Minkowski 3-space, Nonlinear

Anal-Real 2013; 14: 383–396.

[7] Gürbüz N. p-Elastica in the 3-Dimensional Lorentzian Space Forms. Turk J Math 2006; 30: 33-41.

[8] Hasimoto H. A soliton on a vortex …lament. J Fluid Mech 1972; 51: 477–485.

[9] Hasimoto H. Motion of a vortex …lament and its relation to elastica. J Phys Soc Jpn 1971; 31:

293–294.

[10] Nešovic E, Petrovic-Torgašev M, Verstraelen L. Curves in Lorentzian Spaces. B Unione Mat Ital

2005; 8: 685-696.

[11] Ratcli¤e JG. Foundations of Hyperbolic Manifolds. New York, NY, USA: Springer-Verlag, 1994.

notes on magnetic curves in 3d semi-riemannian manifolds

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