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Volume 2012, Issue No.6, Dt. 26 June 2012 Time: 12h55m PM
Artificial Gravity on Space Craft (PART I)

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, nonholonomic system

ABSTRACT

A complete set of new expressions obtained for the system of a space ship under artificial gravity. The Coriolis force 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 are used. The Riemannian metric is explicitly formulated, by giving the relevant matrix formulation. This helped to obtain the several of the General Theory of Relativity curvature tensor field quantities.

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. Professor Bradbury gave an illustration in the book on Theoretical Mechanics in the year 1967 but without the consideration of Lagrange-Euler coupled formulation. 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 mechanical system by using a canonical linear connection on configuration space.

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. [Aurel Bejancu 2007,Bejancu & Farren 2006, Dragovic & Gajic 2003, Bates & Sniatyski 1993, Bloch 2003, Montgomery 2002, Tavares 2003, Vranceanu 1926, and Yano, Kom 1984 to mention a few. Others listed below]. Please see the notation and some other details of the paper by Aurel Bejancu 2007 and our approach radically differs from his methedology and the details of the involved computations.

I consider specifically the nonholonomic constraints, given by the equations

dx\dt+ R cos θ cos φ dφ\dt + R cos φ dψ\dt – R(w+ sin θ sin φ) dθ\dt=0.

dy\dt+ R cos θsin φ dφ\dt + R (sin φ+w) dψ\dt + R sin θ cos φ dθ\dt=0.

with of course, the variables as     
                r¹= φ,     r²=ψ,     r³= θ,       s⁴= x,    s⁵  =y.    

 The coefficient equations of connections taken to be except that we have involved the factor, w to take into account the angular velocity of rotation with just modesty.

 A⁴₁  = +R *cosθ* cosφ, A⁴₂ = +R *cosφ,     A⁴₃  = - R* (w+ sinθ sinφ)

 A⁵₁ = +R *cosθ *sinφ, A⁵₂ = +R * (sinφ+ w), A⁵₃  = + R* sinθ* cosφ.

We expect to have a matrix for the adapted frame field
{ δ/δrα , ∂/∂sa}   given by

[ R^2* (A+ C*cosθ^2)                R^2 * C*cosθ          0                                      -R*cos θ *cos φ        -R*cos θ*cos φ;    

R^2 * C*cos θ                                R^2*(2+w^2)          0                                   -R* cos φ                      -R*(sin φ +w);

0                                                           0              R^2*( sinθ^2 +w^2+A)           R*(w+sin θ sin φ )      -R*(sin θ*cos φ);

-R*cos θ *cos φ                            -R* cos φ    -R*(w+sin θ sin φ )                      R^2 + w^2                             0;




-R*cos θ*cos φ  -R*(sin φ +w)  -R*(sin θ*cos φ)          0          R^2  +  w^2  ]             

Then we have the truncated to get a third order matrix as

[R^2* (A+ C*cosθ^2)                 R^2 * C*cos θ                   0;


R^2 * C*cos θ                             R^2*(2+w^2)                     0;


0                                                  0                    R^2*( sinθ^ 2 +w^2+A)]

Then by putting

B   =   R^2*( - C*cos θ ^ 4+ cosθ ^2*C*w^2 + 2*cos θ^ 2 *C + w^2*A + 2*A)
for the determinant expression we get the inverse matrix as

[ (w^2+2)/B            -cosθ^2/B                 0;


- cosθ^2 /B          (C*cosθ^2+A)/B        0;


        0                            0          1/(R^2*( sinθ^2 + w^2+A))]

We have accordingly the gμν given by,

g41= -R*cosθ *cosφ; g42= -R*cosφ; g43= + R*(w+sin θ sin φ )

g51= -R*cosθ*cosφ; g52= -R*(sin φ+w); g53=-R*sin θ cos φ

The calculation of the nonholonomic Christoffel symbols of second kind has been performed and are listed below. Here {μ: αβ} as the second order Christoffel symbols.

{1; 13}= {1:31} = - (w^2+2)*cosθ*sinθ *(C*R^2)/B + cosθ^2 *C*R^2*sinθ/ (2*B);

{1:23}= {1:32} = - (C*R^2)*(w^2+2)*sinθ/ (2*B) – (C*R^2) * cosθ^ 2*w’ *w/B;

{2:13}= {2:31} =+ (C*R^2)*sinθ*cosθ^3/B – (1/2)* (C*R^2)* (C*cosθ^2+A)* sin θ /B;

{2:23}= {2:32} = +(C*R^2)*sinθ*cosθ^2/ (2*B) + (C*R^2) * (w*w’)* (C*cosθ^ 2+ A)]/B;

{3:21}= {3:12} = +C*sinθ/ (2*(sinθ ^2+w^2+A));

{3:11)= +C*cosθ*sinθ/ (sinθ ^2 +w^2+A);

{3:22}= - C*w*w’/ (sinθ ^2 +w^2+A);

{3:33}= + (sinθ *cosθ + w *w’)/ (sinθ ^2 +w^2+A);

Note that we obtained new expressions for {3:22} and {3:33}.

We have the expressions

B ⁴₁₂  = + R*sinφ;       B⁴₂₁= - R*sinφ;

B⁴₁₃  =+R*δw/δφ ;      B⁴₃₁= - R*δw/δφ ;

Further we have

        B⁴₃₂   =   B⁴₂₃     = 0;

We have the expressions

         B⁵₁₂     = - R^2 *cosφ - R*δw/δφ ;   

         B⁵₂₁      = +R*cosφ + R*δw/δφ

Again we have
        B⁵₁₃= 0 = B⁵₃₁;


Finally

      B⁵₂₃       =  - R* δw/δθ ;

Next, we tackle the equations

B ^a₁₂ g_a1   =    -  B ^a₂₁g_a1  = R^2 *cosθ *sinφ * δw/δφ

B ^a₁₂ g_a2   =    - B ^a₂₁g_a2   = R^2 *( cosφ*w + (sinφ + w) * δw/δφ)

B ^a₁₂ g_a3  =- B ^a₂₁g_a3  =   R^2 *(w*sinφ + sinθ *(1+ cosφ* δw/δφ))

also

B ^a₁₃ g_a1   =    -  B ^a₃₁g_a1    = - R^2 * cosθ * cosφ* δw/δφ

B ^a₁₃ g_a2   =    - B ^a₃₁g_a2    = - R^2 * cosφ * δw/δφ

B ^a₁₃ g_a3     =   - B ^a₃₁g_a3    = + R^2 * δw/δφ * (w + sinθ*sinφ)

further

B ^a₂₃ g_a1   =    -  B ^a₃₂g_a1    = + R^2 * cosθ * sinφ * δw/δθ

B ^a₂₃ g_a2   =    - B ^a₃₂g_a2     = + R^2 * (sinφ + w) * δw/δθ

B ^a₂₃ g_a3     =   - B ^a₃₂g_a3     = + R^2 * sinθ * cos φ * δw/δθ

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

and in addition, other data of this paper would be presented in my next publication.

CONCLUSIONS

I have developed a new approach (Part I) 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. We 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 II and as well 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. Aureal Bejancu, “On the geometry of nonholonomic mechanical systems with vertical   distribution”,
      J. Math. Phys. Vol.48, 052903, 2007


2. Bejancu A and Farran H. R, “Foliations and Geometric Structures”,    Springer, NY, 2006


3. Dragovic V and Gajic B., “The Wagner curvature tensor in nonholonomic mechanics”,
          Regular Chaotic Dyn. Vol.8, 105-124, 2003


4. Bates L and Sniatyski J,       Rep.Math. Phys. Vol.32, 99-115, 1993


5. Bloch A. M, “Nonholonomic Mechanics and Control”, Springer NY, 2003


6. Montgomery R, “A Tour of Subriemanniana Geometries Their Geodesics and Applications”,     Mathematical Surveys Monograph Vol.91 Amer.Math. Soc., Providence R1, 2002


7. Tavares J.N., “About Cartan geometrization of non-holonomic mechanics”,
        J. Geom.Phys Vol.45, 1-23, 2003


8. Vranceanu G, “Sopra le equazioni del moto un sisteme anolonomic”,
     Rend. Accul. Naz. Lincei Vol.4, 508-511, 1926


9. Yano K and Kom M, Structures on Manifolds, World Scientific, Singapore, 1984


10. G. Giachitta, J. Math.Phys, Vol.33(5), 1652-1665, 1992


11. G.Giachetta, L.Mangiarotti and G. Sardanashvily,
      J. Math.Phys, Vol.40(3), 1376-1390, 1999

 

      B⁵₃₂       = + R* δw/δθ;