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Solving the equations of motions of a particles constrained to a sphere.

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a particle moving on a two dimensional sphere of radius r, whose Lagrangian is given by:
    L(q,q')=[itex]\frac{1}{2}[/itex]m[itex]\sum[/itex](qi')2

    (A) Transform the Lagrangian into spherical coordinates and write down the resulting Euler-Lagrange equations.

    (B) Solve the Euler-Lagrange equations corresping to the Lagrangian and write down the solution in the original coordinates (qi). Express your answer to include the integration constants and denote them as the initial position and velocity.

    2. Relevant equations
    [itex]\frac{∂L}{∂q}[/itex]=[itex]\frac{d}{dt}[/itex][itex]\frac{∂L}{∂q'}[/itex]


    3. The attempt at a solution
    I think I have part (A) solved. The problem is part B. I really don't know where to start. But this is what I have for part (A)

    mr2[itex]\phi[/itex]''Sin2(θ)+[itex]\phi[/itex]'θ'2Sin(θ)Cos(θ)=0
    r2[itex]\phi[/itex]'Sin(θ)Cos(θ)=mr2θ''

    I would not know where to start solving these equations.
     
  2. jcsd
  3. Mar 26, 2012 #2
    First of all, you have some typos in your equations, since 2nd equation has 1 time derivative in left and 2 in right side. Also m should drop out everywhere.

    Secondly, you forgot to calculate the last equation (wrt r), which will probably prove to be very useful in solving the set.
     
  4. Mar 27, 2012 #3
    In the second equation I forgot a power of [itex]\phi[/itex]'.
    Also I did not forget to do one of the E.L. equations. Because we are constrained to the surface of a sphere I simply set r'=0 and treated r as a constant parameter instead of a dynamical variable. This reduces the Lagrangian (in spherical coordinates) to a Lagrangian involving two functions instead of three.
    L=[itex]\frac{1}{2}[/itex]m(r2θ'2+r2[itex]\phi[/itex]'2Sin2(θ))

    I also failed to notice that since the derivative of the lagrangian with respect to [itex]\phi[/itex] is zero one of the E.L. equations will yield an integral of motion which should help out a bit.
     
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