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Lagrange multipliers rotating masses connected by spring

  1. Sep 12, 2008 #1
    1. The problem statement, all variables and given/known data

    A particle of mass, m1, is constrained to move in a circle with radius a at z=0 and another particle of mass, m2, moves in a circle of radius b at z=c. For this we wish to write up the Lagrangian introucing the constraints by lagrange multipliers and solve the following equations of motion.

    2. Relevant equations

    Equations of constraint.

    [tex]z1=0\ \ x1^2+y1^2=a^2[/tex]
    [tex]z2=c \ \ x2^2+y2^2=b^2[/tex]

    3. The attempt at a solution

    I am working on the Lagrangian given by

    [tex]L=T-V+\lambda_1\left(-z_1\right)+\lambda_2\left(c-z_2\right)+\lambda_3\left(a^2-x_1^2-y_1^2\right)+\lambda_4\left(b^2-x_2^2-y_2^2\right)[/tex]

    with
    [tex]T=\frac{1}{2}m_1\left(\dot{x}_1^2+\dot{y}_1^2+\dot{z}_2^2\right)+\frac{1}{2}m_2\left(\dot{x}_2^2+\dot{y}_2^2+\dot{z}_2^2\right)[/tex]
    and
    [tex]V=\frac{1}{2}k\left(\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2\right)[/tex]

    From this we get the equations of motion:

    [tex]m_1\ddot{x_1}=-k\left(x_2-x_1\right)-2\lambda_3x_1[/tex]

    [tex]m_1\ddot{y_1}=-k\left(y_2-y_1\right)-2\lambda_3y_1[/tex]

    [tex]m_1\ddot{z_1}=-k\left(z_2-z_1\right)-\lambda_1[/tex]

    [tex]m_2\ddot{x_2}=k\left(x_2-x_1\right)-2\lambda_4x_2[/tex]

    [tex]m_2\ddot{y_2}=k\left(y_2-y_1\right)-2\lambda_4y_2[/tex]

    [tex]m_2\ddot{z_2}=k\left(z_2-z_1\right)-\lambda_2 [\tex]


    Anyone know how to solve these equations of motion
     
  2. jcsd
  3. Sep 14, 2008 #2
    You could reduce the number of coordinates by introducing polar coordinates for each mass. There would be less equations to solve.
     
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