trusciencetrutechnology@blogspot.com
Volume 2012, Issue No.7, Dt. Saturday July 7, 2012 Time: 11h18m.A.M
Artificial Gravity on Space Craft (PART II)
by
Professor Dr Kotcherlakota Lakshmi Narayana
{Retd. Prof. of Phys, SU} 17-11-10, Narasimha Ashram, Official Colony,
Maharanipeta. P. O, Visakhapatnam-530002.
Mobile No: 9491902867

Key words: Artificial Gravity, Space Craft.

ABSTRACT

A complete set of new expressions obtained for the system of a space ship under Artificial Gravity. This forms the paper No.II. The Coriolis force described by the symbol w is specifically considered. A nonholonomic mechanical system of the spacecraft considered to initiate a study of the geometry using conventional Lagrange-Euler equations. The nonholonomic Christoffel symbols of second kind obtained explicitly. The nonzero components of the curvature of Ehresmann connection evaluated.

INTRODUCTION

The spinning the space ship about an axis gives rise to an Artificial Gravity. A flat geometry with an axis pointed upward in the centre would be an ideal example. The centrifugal acceleration results in producing Artificial Gravity. An important quantity considered was the Coriolis Effect. I have added a vertical dimension to serve the purpose of an orientation of the spacecraft. In this paper, I present the details of a nonholonomic and nonsymmetrical symbols that explicitly evaluated.

MODEL

Deliberately I have chosen a model to bring out the salient features of the Artificial Gravity and developed Lagrange-Euler theory for nonholonomic mechanical system with nonsymmetry. This is radical departure from the earlier theories based nonholonomic systems but with involved symmetry considerations. Relevant references in this context are the papers published by several authors in recent times. [lakshminarayana.kotcherlakota@blogspot.com, Artificial Gravity on Space Craft (PART I) June 2012 ,Aurel Bejancu 2007,Bejancu & Farren 2006, Dragovic & Gajic 2003, Bates & Sniatyski 1993, Bloch 2003, Montgomery 2002, Tavares 2003, Vranceanu 1926, and Yano, Kom 1984 and many other references]

A reference to the B coefficients explicitly presented in Paper I, may be referred to carry out the calculations of Paper II.

Γ _α ^γ _β  =   {γ : αβ} + ½ * g^γμ * { B ^a _μα  g_aβ  + B ^a _μβ  g_aα -  B ^a _αβ  g_aμ}

Here we have adopted the symbol {γ : αβ} as a second order Christoffel symbol as we did earlier.

It is possible to deduce the expressions, for example, like these

Γ ₁ ¹ ₁  =   {1 : 11} + ½ * g^1μ * {2* B^a _μ1  g_a1  -  B^a₁₁  g_aμ}

Γ ₁ ¹ ₂ =   {1 : 12} + ½ * g^1μ * { B^a _μ1  g_a2  + B^a_μ2  g_a1 -  B^a₁₂  g_aμ}

Γ ₁ ¹ ₃  =   {1 : 13} + ½ * g^1μ * { B^a _μ1  g_a3  + B^a_μ3  g_a1 -  B^a₁₃  g_aμ}

Γ ₂ ¹ ₁ =   {1 : 21} + ½ * g^1μ * { B^a _μ2  g_a1  + B^a_μ1  g_a2 -  B^a₂₁  g_aμ}

Γ ₂ ¹ ₂   =   {1 : 22} + ½ * g^1μ * {2* B^a _μ2  g_a2  -  B^a₂₂  g_aμ}

Γ ₂ ¹ ₃ =   {1 : 23} + ½ * g^1μ * { B^a _μ2  g_a3  + B^a_μ2  g_a3 -  B^a₂₃  g_aμ}

Γ ₃ ¹ ₁ =   {1 : 31} + ½ * g^1μ * { B^a _μ3  g_a1  + B^a_μ1  g_a3 -  B^a₃₁  g_aμ}

Γ ₃ ¹ ₂ =   {1 : 32} + ½ * g^1μ * { B^a _μ3  g_a2  + B^a_μ2  g_a3 -  B^a₃₂  g_aμ}

Γ ₃ ¹ ₃ =   {1 : 33} + ½ * g^1μ * {2* B^a _μ3  g_a3  -  B^a₃₃  g_aμ}

In addition, expressions for the remaining 18 quantities of   Γ _α ^γ _β    easily obtained. The equations of motion using the variables chosen in this paper obtained using the above stated expressions.

The expression for the  θ- equation given below

θ̈  +    Γ ₁ ³ ₁ * φ̇ ²    +   Γ ₂ ³ ₂  * ψ̇ ²    +    Γ ₃ ³ ₃  * θ̇²     

+  (Γ ₁ ³ ₂  + Γ ₂ ³ ₁) *φ̇ * ψ̇   +  (Γ ₁ ³ ₃  + Γ ₃ ³ ₁)  *φ̇ * θ̇

+    (Γ ₂ ³ ₃  + Γ ₃ ³ ₂ )   * ψ̇ * θ̇  = 0

Similar expressions be written easily for the remaining two equations for the ψ ̈ and φ ̈ respectively.

Here we have  

Γ ₁ ³ ₁  = [C*cos θ*sin θ + δw/δφ*cos θ *cosφ]/ (sin² θ  + w ² + A).

Γ ₂ ³ ₂ = [-C*w*w’ -  δw/δθ *(sinφ + w)]/ (sin² θ  + w ² + A).

Γ ₃ ³ ₃  =[ sin θ cos θ + w* w’]/ (sin² θ  + w ² + A).

Γ ₁ ³ ₂  + Γ ₂ ³ ₁  = {C*sinθ + [δw/δφ *cosφ - δw/δθ * cos θ*sinφ] / (sin² θ  + w ² + A).

Take now that the expressions δw/δθ=0; δw/δφ=0;

As a result (Γ ₁ ³ ₃  + Γ ₃ ³ ₁)=0 and (Γ ₂ ³ ₃  + Γ ₃ ³ ₂ )=0;

Note we have Γ ₃ ³ ₂=0;

The θ- equation simplifies to

 θ̈   + C*cos θ*sin θ/(sin² θ  + w ² + A)  * φ̇ ² 

 

    - C*w*w’ (sin² θ  + w ² + A) * ψ̇ ² 

+ [ sin θ* cos θ + w* w’]/(sin² θ  + w ² + A)   * θ̇² +


C*sinθ/(sin² θ  + w ² + A)   *φ̇ * ψ̇  =  0

Extremely simple equation of consideration is

θ̈  + [ sinθ *cos θ + w* w’]/ (sin² θ  + w ² + A) * θ̇²  = 0

 if the sum of the remaining expressions turn out to vanish.

This equation solved with the attended Physics.


The next equation I discuss is the one for variable ψ



 

Ψ̈  + Γ ₁ ² ₁  * φ̇ ² +  Γ ₂ ² ₂  * ψ̇ ²  +  Γ ₃ ² ₃  * θ̇² +

(Γ ₁ ² ₂  + Γ ₂ ² ₁) *φ̇ * ψ̇         +      (Γ ₁ ² ₃  + Γ ₃ ² ₁) *φ̇ * θ̇ 

+   (Γ ₂ ² ₃  + Γ ₃ ² ₂ ) * ψ̇ * θ̇   =  0;

This equation seems to be somewhat associated with the equation

for φ since the expression for Γ ₁ ¹ ₁ vanishes if assume that  δw/δφ=0;

It is worthwhile to examine these two equations together instead of

individually and hence is postponed for the present.

 

CONCLUSIONS

I have developed an approach (Part II) of the Lagrange-Euler equations to make them readily available for gravitational considerations of space travel and objects like the ones, as the several space ships and people dwelling in them. I expect that this paper would be a useful tool to study the nonholonomic and as well nonsymmetrical systems of importance to Astrophysics and the Planetary Travel ships. I wish to publish in due course of time the PART III.


ACKNOWLEDGMENT
I am indebted to Late Prof. K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) of Andhra University, (Waltair) Visakhapatnam for initiating me to do research studies and for his constant guidance during my professional career. His gesture of acquiring books from Moore Market in Madras in 1956 Deepavali vacation has helped me a lot to gain some fundamental knowledge in Mathematical Physics. The books purchased for me Introductory Calculus, Two Volumes by Courant, Pure Mathematics, by G H Hardy, Theory of Relativity by R C Tolman and Modern Algebra by Barnard etc., have helped me a lot. With his kind gesture, I could also attend the International Seminar on Education and Research held in University of Uppsala, Sweden during the year 1964-1965 where I could meet learned Professors.

REFERENCES


1. lakshminarayana.kotcherlakota@blogspot.com, Volume 2012, Issue No.6, Dt. 26 June 2012 Time: 12h55m PM


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