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Lagrangian mechanics

  1. Nov 21, 2006 #1
    I have two point masses that are connected by a massless inextensible rope of length [tex]l[/tex] which passes through a small hole in a horizontal plane. The first point mass moves without friction on the plane while the second point mass oscillates like a simple pendulum in a constant gravitational field of strength [tex]g[/tex].

    I have used three constraints whereas one is [tex]l_1 + l_2 = l[/tex] and found three equations of motion:

    [tex]2 \dot{l_1} \dot{\theta_1} + l_1 \ddot{\theta_1} = 0[/tex]

    [tex](l - l_1)\ddot{\theta_2} - 2 \dot{l_1} \dot{\theta_2} + g \sin{\theta_2} = 0[/tex]

    [tex](m_1 + m_2) \ddot{l_1} - m_1 l_1 \dot{\theta_1}^2 + m_2 ((l - l_1) \dot{\theta_2}^2 + g \cos{\theta_2}) = 0[/tex]

    First, can this be correct? Second, how do I solve these?

    I have also found one conserved quantity,

    [tex]\frac{\partial L}{\partial \theta_1} = 0[/tex]

    wich says that the generalized momentum is conserved. How can I find another conserved quantity?
     
  2. jcsd
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