Solving Lagrangian Mechanics for 2 Point Masses Connected by a Rope

• Logarythmic
In summary, the conversation discusses a system of two point masses connected by a massless rope and constrained by a horizontal plane and gravity. The equations of motion have been derived and a conserved quantity has been found. To solve the equations, techniques from classical mechanics and Lagrangian mechanics can be used, and another conserved quantity can be found using the properties of the Lagrangian and symmetry arguments.
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?

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!

1. What is Lagrangian Mechanics?

Lagrangian Mechanics is a branch of classical mechanics that provides a mathematical framework for analyzing the motion of systems in terms of generalized coordinates and their corresponding momenta. It is named after the Italian mathematician and astronomer Joseph-Louis Lagrange.

2. How do you solve Lagrangian Mechanics for 2 point masses connected by a rope?

To solve Lagrangian Mechanics for 2 point masses connected by a rope, you must first determine the kinetic and potential energies of the system, using the generalized coordinates and their corresponding momenta. Then, you can use the Euler-Lagrange equations to find the equations of motion for the system. Finally, you can solve these equations to determine the position, velocity, and acceleration of the masses at any given time.

3. What are generalized coordinates and momenta?

Generalized coordinates are a set of variables that describe the configuration of a system in terms of its degrees of freedom. They can be chosen to simplify the analysis of the system. Momenta, on the other hand, are the conjugate variables to the generalized coordinates and represent the momentum of the system in each degree of freedom.

4. Can Lagrangian Mechanics be used for any type of system?

Yes, Lagrangian Mechanics can be used for any type of system, as long as the system can be described in terms of generalized coordinates and their corresponding momenta. It is particularly useful for systems with complex or non-Cartesian coordinates, such as systems with constraints or systems in rotating frames of reference.

5. What are the advantages of using Lagrangian Mechanics over Newtonian Mechanics?

Lagrangian Mechanics offers several advantages over Newtonian Mechanics. It provides a more elegant and concise formulation of the equations of motion, which can simplify the analysis of complex systems. It also takes into account constraints and allows for a more systematic approach to solving problems. Additionally, it is more suitable for systems with non-Cartesian coordinates and is often used in advanced physics and engineering applications.

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