A modified Fermi-Walker derivative for inextensible flows of binormal spherical image

Gülden Altay Suroğlu

doi:10.1515/phys-2018-0003

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A modified Fermi-Walker derivative for inextensible flows of binormal spherical image

Gülden Altay Suroğlu

Published/Copyright: March 8, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

Fermi-Walker derivative and biharmonic particle play an important role in skillful applications. We obtain a new characterization on binormal spherical indicatrix by using the Fermi-Walker derivative and parallelism in space. We suggest that an inextensible flow is the necessary and sufficient condition for this particle. Finally, we give some characterizations for a non-rotating frame of this binormal spherical indicatrix.

Keywords: Fermi-Walker derivative-parallelism; Energy; Bienergy; Heisenberg group; Faraday tensor

PACS: 02.40.Ma; 02.30.Jr; 02.90.+p

1 Introduction

In the literature there are many studies on Fermi-Walker transport and Fermi-Walker derivative. Simple description for the construction of Fermi-Walker transported frames out of an arbitrary set of tetrad fields was presented in Ref. [1, 2, 3].

Recently, a research on Fermi-Walker transports has been expanded to Minkowski spacetime in Ref. [4, 5]. Frenet-Serret equations constructed by Synge on world lines are a strong instrument for studying motion of non-zero rest mass for test particles in an assumed gravitational field [6, 7, 8, 9, 10, 11]. Also, Frenet-Serret equations have been generalized from non-null to null trajectories in a spacetime by using a new formalism with Fermi-Walker transport in Ref. [4].

In Ref. [5, 10, 11, 12, 13, 14, 15, 16], some curves corresponding to their flows has been investigated. A new characterization of inextensible flows for curves with Fermi-Walker derivative and its parallelism on the 3-dimensional space has also been constructed. More precisely, they have constructed new figures as illustrations of the moving charged particle in electromagnetic field. Flows of curves of a given curve are also widely studied in Ref. [17, 18]. Some characterizations of curves and surfaces are given in Ref. [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

In Ref. [2], Fermi-Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi-Walker terms with Darboux vector are given in Minkowski 3- dimensional space. In Ref. [15], flows of biharmonic particles on a new spacetime are defined by using Bianchi type-I (B-I) cosmological model. A geometrical description of timelike biharmonic particle in spacetime is also given. In Ref. [17], a new method for inextensible flows of timelike curves in a conformally flat, quasi conformally flat and conformally symmetric 4-dimensional LP-Sasakian manifold is developed.

The structure of the paper is as follows. First, we construct a new characterization for inextensible flows of binormal spherical indicatrix and Fermi-Walker parallelism by using Fermi-Walker derivative in space. Finally, we give some characterizations for non-rotating frame of binormal spherical image.

2 Preliminaries

In this section, we study relationship between the Fermi-Walker derivative and the Frenet fields of curves. Moreover, we obtain some characterizations and an example of the curve.

Fermi transport and derivative have the following theories.

Fermi-Walker transport is defined by

∇TFWV=∇TV−TV,∇TT+∇TTV,T=0.(1)

∇TFWV is called Fermi-Walker derivative of V with respect to T along the curve in space.

With this definition the following features are satisfied [24]:

If the curve is a geodesic, then the Fermi-Walker transport is identical to parallel transport: if ∇_TT = 0, then ∇TFWV = ∇_TV.
∇TFWT = 0, that is, the tangent to the curve is always Fermi-Walker transported.
V and W are Fermi-Walker transported vector fields, their inner product remains constant along the curve

ddsV(s),W(s)=∇TV,W+V,∇TW,(2)

∇TFWV=0=∇TV−TV,∇TT+∇TTV,T.(3)

If we write the equation (3) at equation (2)

ddsV(s),W(s)=TV,∇TT−∇TTV,T,W+V,TW,∇TT−∇TTW,T=T,WV,∇TT−∇TT,WV,T+V,TW,∇TT−V,∇TTW,T=0.(4)

is attained, [25].

Physical sense of above attribute is that V is orthogonal to T along curve. Thus, horizontal change in it along the curve can only stem from rotation of the vector in a plane perpendicular to T. Property of Fermi-Walker transport ∇TFWV = 0 intends that vector is moved without any rotation. This is fundamental to detect the action of gyroscopes when moved with accelerated observers.

3 Construction of Fermi-Walker derivative for inextensible flows of curves

Lemma 3.1

Let α : I ⊂ R → 𝕄 be a curve in space and V be a vector field along the curve α. For a map Γ : I × (–ε, ε )→ 𝕄, putting Γ (s, 0) = α (s), (∂Γ∂s(s,t))=V(s).

Therefore, the following functions can be obtained:

Speed function v(s, t) = ∂Γ∂s(s,t),
Curvature function κ (s, t) of α_t(s),
Torsion function τ (s, t) of α_t(s).

The variations of those functions at t = 0 are

V(v)=∂v∂t(s,t)|t=0=g(∇TV,T)v,(5)

V(κ)=∂κ∂t(s,t)|t=0=g(∇T2V,N)−2κg(∇TV,T)+g(R(V,T)T,N),(6)

V(τ)=∂τ∂t(s,t)|t=0=1κg(∇T2V+R(V,T)T,Bs+κ(∇TV,B)+τg(∇TV,T)+g(R(V,T)N,B),(7)

where R is the curvature tensor.

Second, a flow of α_t(s) may be represented as

∂αt∂t=β1T+β2N+β3B,(8)

where β₁, β₂, β₃ are smooth functions.

On the other hand, Körpınar–Turhan obtained flows of binormal spherical images of curves in space [7].

Now, we investigate conditions of Frenet vectors.

Theorem 3.2

Ifϕis binormal spherical indicatrix ofα, then

∇tTϕ=−∇tN=β1κ−β3τ+∂β2∂sT−CB,(9)

∇tNϕ=∂∂tκβ+β2τ+∂β3∂sτβT+Cτβ+β1κ−β3τ+∂β2∂sκβN+β2τ+∂β3∂sκβ−∂∂tτ βB,(10)

∇tBϕ=∂∂tτβ−β2τ+∂β3∂sκβT+β1κ−β3τ+∂β2∂sτβ−CκβN+∂∂tκβ+β2τ+∂β3∂sτβB,(11)

where

β=κ2+τ2,C=∇tN,B.(12)

Theorem 3.3 (Main Theorem)

∇t∇sFWX=∇sFW∇tX+∇tκϕ(Bϕ∧X)+κϕ(∇tBϕ∧X).(13)

Proof

By using the definition of Fermi-Walker transport we have the above equality. This completes the proof.

Theorem 3.4

∇sFW∇tTϕ=∂∂sβ1κ−β3τ+∂β2∂sT+[κβ1κ−β3τ+∂β2∂s+Cτ−κϕτβC+κββ1κ−β3τ+∂β2∂s]N−∂C∂sB.(14)

Proof

From Serret-Frenet formulas and Fermi-Walker derivative, we have

∇sFWX=∇sX−κϕ(Bϕ∧X)(3.11)

Using flow of T^ϕ, we obtain

∇sFW∇tTϕ=∇s∇tTϕ−κϕ(B∧∇tTϕ).(3.12)

Since the above equation, we get

∇s∇tTϕ=∂∂sβ1κ−β3τ+∂β2∂sT−∂C∂sB+κβ1κ−β3τ+∂β2∂s+CτN−κB∧∇tTϕ.(15)

By using the properties of cross product we can easily write that

Bϕ∧∇tTϕ=τβT+κβB∧β1κ−β3τ+∂β2∂sT−CB=τβC+κββ1κ−β3τ+∂β2∂sB∧T=τβC+κββ1κ−β3τ+∂β2∂sN.(16)

Then it is easy to obtain that

∇sFW∇tTϕ=∂∂sβ1κ−β3τ+∂β2∂sT+[κβ1κ−β3τ+∂β2∂s+Cτ−κϕτβC+κββ1κ−β3τ+∂β2∂s]N−∂C∂sB,(17)

which completes the proof.

Theorem 3.5

∇sFW∇tNϕ=[∂∂s∂∂tκβ+β2τ+∂β3∂sτβ−κCτβ+β1κ−β3τ+∂β2∂sκβ−κκϕβCτβ+β1κ−β3τ+∂β2∂sκβ]T+[κ∂∂tκβ+β2τ+∂β3∂sτβ+∂∂sCτβ+β1κ−β3τ+∂β2∂sκβ−τβ2τ+∂β3∂sκβ−∂∂tτβ+[κκϕβ∂∂tκβ+β2τ+∂β3∂sτβ−τββ2τ+∂β3∂sκβ−∂∂tτ β]]N+[∂∂sβ2τ+∂β3∂sκβ−∂∂tτβ+τCτβ+β1κ−β3τ+∂β2∂sκβ+κϕτβCτβ+β1κ−β3τ+∂β2∂sκβ]B.(18)

Proof

This can be verified using an argument similar to above theorem.

Theorem 3.6

∇sFW∇tBϕ=[∂∂s∂∂tτβ−β2τ+∂β3∂sκβ−κβ1κ−β3τ+∂β2∂sτβ−Cκβ−κκϕββ1κ−β3τ+∂β2∂sτβ−Cκβ]T+[κ∂∂tτβ−β2τ+∂β3∂sκβ+∂∂sβ1κ−β3τ+∂β2∂sτβ−Cκβ−τ∂∂tκβ+β2τ+∂β3∂sτβ+κϕ[κβ∂∂tτβ−β2τ+∂β3∂sκβ−τβ∂∂tκβ+β2τ+∂β3∂sτβ]]N+[∂∂s∂∂tκβ+β2τ+∂β3∂sτβ+τβ1κ−β3τ+∂β2∂sτβ−Cκβ+τκϕββ1κ−β3τ+∂β2∂sτβ−Cκβ]B.(19)

Proof

This can be verified using an argument similar to above theorem.

Using above theorems, we get the following corollaries by straight-forward computations.

Corollary 3.7

If ∇_tT^ϕalong the curve, parallel to the Fermi–Walker terms, then

∂∂sβ1κ−β3τ+∂β2∂s=0,(20)

κβ1κ−β3τ+∂β2∂s+Cτ−κϕτβC+κββ1κ−β3τ+∂β2∂s=0,(21)

∂C∂s=0.(22)

Corollary 3.8

If ∇_tN^ϕalong the curve, parallel to the Fermi–Walker terms, then

[∂∂s∂∂tκβ+β2τ+∂β3∂sτβ−κCτβ+β1κ−β3τ+∂β2∂sκβ−κκϕβCτβ+β1κ−β3τ+∂β2∂sκβ]=0,[κ∂∂tκβ+β2τ+∂β3∂sτβ+∂∂sCτβ+β1κ−β3τ+∂β2∂sκβ−τβ2τ+∂β3∂sκβ−∂∂tτβ+[κκϕβ∂∂tκβ+β2τ+∂β3∂sτβ−τββ2τ+∂β3∂sκβ−∂∂tτ β]]=0,[∂∂sβ2τ+∂β3∂sκβ−∂∂tτβ+τCτβ+β1κ−β3τ+∂β2∂sκβ+τκϕβCτβ+β1κ−β3τ+∂β2∂sκβ]=0.(23)

Corollary 3.9

If ∇_tB^ϕalong the curve, parallel to the Fermi–Walker terms, then

[∂∂s∂∂tτβ−β2τ+∂β3∂sκβ−κβ1κ−β3τ+∂β2∂sτβ−Cκβ−κκϕββ1κ−β3τ+∂β2∂sτβ−Cκβ]=0,[κ∂∂tτβ−β2τ+∂β3∂sκβ+∂∂sβ1κ−β3τ+∂β2∂sτβ−Cκβ−τ∂∂tκβ+β2τ+∂β3∂sτβ+κϕ[κβ∂∂tτβ−β2τ+∂β3∂sκβ−τβ∂∂tκβ+β2τ+∂β3∂sτβ]]=0,+[∂∂s∂∂tκβ+β2τ+∂β3∂sτβ+τβ1κ−β3τ+∂β2∂sτβ−Cκβ+τκϕββ1κ−β3τ+∂β2∂sτβ−Cκβ]=0.(24)

4 Applications to electrodynamics

In this section, we construct the Fermi–Walker derivative in the motion of a charged particle under the action of only electric or magnetic fields.

The equation of motion of a charged particle of mass m and electric charge q under the electric field 𝓔 and magnetic field 𝓑 is given by the Lorentz equation. In Gaussian system of units, we have [25]:

mdvds=qE+qv×B.(25)

Only a magnetic induction 𝓑 (no electric field 𝓔), the equation of motion is
mdTϕds=qTϕ×B.(26)
From above equation and Frenet frame, we easily choose
B=−κϕmqBϕ.(27)
Then,
B=−κϕmqτβT+κβB.(28)
Therefore, we can write
∇sFWB=−∂∂sκϕmqτβT−κϕmqτβκN+κϕmqκτβ−∂∂sκϕmqκβB.(29)
which implies that
∂∂sκϕmqτβ=0,(30)
κϕmqτβκ=0,(31)
κϕmqκτβ=∂∂sκϕmqκβ.(32)
On the other hand, we obtain
∇tB=−∂∂tκϕmqτβ−κϕmq∂∂tτβ−β2τ+∂β3∂sκβT−κϕmqβ1κ−β3τ+∂β2∂sτβ−CκβN+κβ−κϕmq∂∂tκβ+β2τ+∂β3∂sτβB.(33)
If ∇_t𝓑 along the curve, parallel to the Fermi–Walker terms, then
−∂∂tκϕmqτβ−κϕmq∂∂tτβ−β2τ+∂β3∂sκβ=0,(34)
κϕmqβ1κ−β3τ+∂β2∂sτβ−Cκβ=0,(35)
κβ−κϕmq∂∂tκβ+β2τ+∂β3∂sτβ=0.(36)
Only an electric induction 𝓔 (no magnetic field 𝓑), the equation of motion is
E=mκϕqNϕ.(37)
Then, we easily have
∇sFWE=∂∂smκϕqκβ+mκϕτϕqτβT+mκϕτϕqκβ−∂∂smκϕqτβB,(38)
which implies that
∂∂smκϕqκβ+mκϕτϕqτβ=0,(39)
Figure 1
Binormal indicatrix of time helix is illustrated with Magenta, Cyan, Green color at the time t = 1.2, t = 1.8, t = 2.2, respectively
mκϕτϕqκβ−∂∂smκϕqτβ=0.(40)
On the other hand, we obtain
∇tE=∂∂tmκϕqκβ+∂∂tκβ+β2τ+∂β3∂sτβT+Cτβ+β1κ−β3τ+∂β2∂sκβN+−∂∂tmκϕqτβ+β2τ+∂β3∂sκβ−∂∂tτ βB.(41)
If ∇_t𝓔 along the curve, parallel to the Fermi–Walker terms, then
κβ−κϕmq∂∂tκβ+β2τ+∂β3∂sτβ=0,(42)
Cτβ+β1κ−β3τ+∂β2∂sκβ=0,(43)
−∂∂tmκϕqτβ+β2τ+∂β3∂sκβ−∂∂tτ β=0.(44)

5 Some pictures

In this section we draw some pictures corresponding to different cases by using a following example:

The time helix is parametrized by

αts=(Atcoss,Atsins,Bts),(45)

where A, B are functions of time only.

6 Conclusions

Fermi-Walker transport and inextensible flows play an important role in geometric design and theorical physics.

In this paper, we have studied Fermi-Walker derivative and Fermi–Walker parallelism for binormal indicatrix. The aim of this work is to show inextensible flows of Fermi-Walker derivative by using curvatures of curves.

Furthermore, using the Frenet frame of the given curve, we present some partial differential equations. We have given some illustrations together with some examples, which we have used flows of Frenet frame and Fermi derivative in space. Finally, we construct the Fermi–Walker derivative in the motion of a charged particle.

In our future work under this theme, we propose to study the conditions on the Fermi-Walker derivative and Fermi-Walker parallelism for spherical indicatrix in Minkowski space.

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Received: 2016-08-08

Accepted: 2017-11-24

Published Online: 2018-03-08

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Keywords for this article

Fermi-Walker derivative-parallelism; Energy; Bienergy; Heisenberg group; Faraday tensor

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