# Lagrangian mechanics

1. Nov 21, 2006

### Logarythmic

I have two point masses that are connected by a massless inextensible rope of length $$l$$ 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 $$g$$.

I have used three constraints whereas one is $$l_1 + l_2 = l$$ and found three equations of motion:

$$2 \dot{l_1} \dot{\theta_1} + l_1 \ddot{\theta_1} = 0$$

$$(l - l_1)\ddot{\theta_2} - 2 \dot{l_1} \dot{\theta_2} + g \sin{\theta_2} = 0$$

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

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

I have also found one conserved quantity,

$$\frac{\partial L}{\partial \theta_1} = 0$$

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