Solving Lagrangian Mechanics for 2 Point Masses Connected by a Rope

[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?

 

[Jhero]

Hello,

Thank you for sharing your equations and question. Your equations do seem to be correct for the system you described. To solve these equations, you will need to use techniques from classical mechanics, specifically Lagrangian mechanics. You can start by finding the Lagrangian of the system, which is the difference between the kinetic and potential energy. Then, you can use the Euler-Lagrange equations to find the equations of motion for each variable.

To find another conserved quantity, you can use the fact that the Lagrangian is independent of time. This means that the total energy of the system, which is the sum of the kinetic and potential energy, will be conserved. You can also use symmetry arguments to find other conserved quantities, such as the angular momentum.

I hope this helps and good luck with your calculations!