Gantmacher’s Formula

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https://doi.org/10.22214/ijraset.2020.32662

December 2020

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 8 Issue XII Dec 2020- Available at www.ijraset.com

Gantmacher’s Formula Ram Swaroop Meghwal1, DeepMala Meena2 1, 2

Dept. of Physics, Govt. College, Kota (Raj.)-INDIA.

Abstract: In theory of differential equations we learn to solve (dy/dx)+Py=Q. But Its vector analogue as encountered in case of Lorentz force is extremely tedious and firstly, it was done by Gantmakher. Here a simplified version is presented for scholarly attention. Keywords: Gantmacher, vector differential equation, Lorentz force problem. I. INTRODUCTION General principles of dynamics are either formulated in differential or in integral forms [1]. The differential or integral forms are many times of implicit nature in such a way that solution appears in terms of solution itself [2]. When such form acquire the vector attribute complexity aggravates [3]. Lorentz force [4] is one of such complexities.

 d 2r m e dt 2

II. STATEMENT OF THE PROBLEM AND ITS SOLUTION

         eB dv eE   dr  E  dt  B   dt  m  v  m  g  v    

The cross product can be made to produce two components only if one of the vectors is single component vector. Now velocity vector is not a constant vector, hence we may choose ω to be a single component vector [5]. Moreover, we must allow v to have all three components. There will be no loss of generality if g is confined to x-z plane.

    zˆ   0,0,  rectangular   ,0, 0 spherical  g  g x xˆ  g z zˆ   g x ,0, g z rectangular   g , ,0 spherical  v  v x xˆ  v y yˆ  v z zˆ  v x ,v y ,v z   v ,  ,  spherical rectangular

xˆ yˆ zˆ    v    v x v y v z  v y xˆ  v x yˆ  ˆ  v 0  v 0 y xˆ  v 0 x yˆ 0

0



   ˆ  ˆ  v 0   v 0 x xˆ  v 0 y yˆ,  ˆ  g  g x yˆ,  ˆ   ˆ  g   g x xˆ

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 8 Issue XII Dec 2020- Available at www.ijraset.com

 dv    dv x  g v     g x  v y dt dt dv y  v x dt dv z  gz dt

...(1) ...(2) ...(3)

Differentiating Eq. (1) and using Eq. (2) , we obtain

dv y d 2v x dgx d 2v x 2     0   (   v )    vx  0 x 2 2 dt dt dt dt v x  A cos t  B sint , v 0x  A cos0o  B sin0o  A v x v 0x cos t  B sint dv x gx 1 d  gx  v y v y    v 0x cos t  B sint   dt   dt g g v y   x v 0x sint  B cos t , v 0y   x  B   v x v 0x cos t v 0y sint  vy  

gx sint 

gx g  v 0x sint  v 0y  x   

  (4)

  cos t 

gx v y  v 0x sint v 0y cos t   cos t  1 

  (5)

v z v 0z  gzt

  (6)

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 8 Issue XII Dec 2020- Available at www.ijraset.com Further results will be obtained after integrating Eqs. (4)-(6)

g dx  v 0 x cos t  v 0 y sin t  x sin t dt  sin t  cos t g x  cos t  x  v 0x . v 0y .  . c     v g 1 g 1  x 0  v 0y .  x .  c  c  x 0  0y  x2 ,      sin t 1  cos t g x 1  cos t  x  x 0  v 0x .  v 0y .  .     2

sin t  t  t 1  cos t g x 1  cos t  12 t   12 t   x  x 0 v 0x . v 0y .  .     1  sin t  t  x  x 0  v 0x t  g x t 2 v 0x  2  

  cos t  1    v 0y    gx    

2

 cos t  1  1 t 2  2   2     

g dy  v 0 x sin t  v 0 y cos t  x  cos t  1 dt  cos t sin t g x  sin t   y v 0 x  v 0y   t D         v 0x v  D  D  y 0  0x ,   cos t  1 sin t  t  t  sin t  t   y  y 0 v 0 x  v 0y  gx     2   cos t  1 sin t  t  sin t  t   y  y 0 v 0y t  v 0x  v 0y  gx     2    y0 

g x   sin t  t   cos t  1    y  y 0  v 0y t  v 0x   v    0y            dz t2  v 0z  g z t  z  z 0  v 0z t  g z dt 2

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 8 Issue XII Dec 2020- Available at www.ijraset.com

 1  sin t  t   cos t  1  ˆ  g x xt ˆ 2  v 0 x xˆ  ˆ r  x 0 xˆ  v 0 x xt  v x  0y   2        cos t  1  1 t 2   cos t  1  2  ˆ ˆ ˆ  g x xˆ   y y  v yt  v y 0 0y 0x      2     g x   sin t  t   t2 ˆ  g z zˆ  v 0 y yˆ  yˆ    z 0 zˆ  v 0 z zt     2     1   cos t  1   r0  v 0t  gt 2   v 0 y xˆ  v 0 x yˆ  2     cos t  1  1 t 2  gx  sin t  t   2   g x xˆ    yˆ  v 0 x xˆ  v 0 y yˆ   2              1   cos t  1   r0  v 0t  gt 2     v ˆ  0  2     cos t  1  1 t 2   2       g ˆ ˆ      2     ˆ  g   sin t  t    ˆ v 0     ˆ       





  1   cos t  1     r0  v 0t  gt 2      v 0  2 2    cos t  1  1 t 2     2         g    4      sin t  t        g      v 0     3  

    dr          dv  g   v  gt    r  c  v 0    r 0  c dt dt        v  v 0  gt     r  r0 

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 8 Issue XII Dec 2020- Available at www.ijraset.com III. DISCUSSION The problem involving vector nature of differential equation is dealt by reducing it to a planar structure. Then solution is obtained in somewhat easier way. Again combining the solutions of two components of planar domain, we get the general solution. This is the simplest way of getting Gantmacher’s formula. REFERENCES [1] [2] [3] [4] [5]

F R Gantmacher, Lectures of analytical mechanics, Mir 1975 Griffiths, David J. Introduction to electrodynamics ,: Prentice-Hall. 1999. Huray, Paul G. . Maxwell's Equations. Wiley-IEEE.2010 J.H.Field, Am. J. Phys. 69 569 , 2001. H.Goldstein, ‘Classical Mechanics’, Addison-Wesley, Reading Massachusetts, 1959

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