# Lagrangian Problem. Two masses on a massless circle

• Xyius
In summary, the problem involves two equal masses attached to a massless hoop of radius R that is free to rotate in a vertical plane. The angle between the masses is 2*theta and the goal is to find the frequency of small oscillations. Using the Lagrangian method, the equation \ddot{\theta}-\frac{5g}{9r^2}\theta=0 is obtained, which suggests that g may be negative in the chosen coordinate system. A minor algebra error may have caused confusion, but the problem can be solved by ensuring consistency in the chosen coordinate system.

## Homework Statement

Two equal masses are glued to a massless hoop of radius R that is free to rotate about its center in a vertical plane. The angle between the masses is 2*theta. Find the frequency of small oscillations.

## Homework Equations

$$\frac{d}{dt} \frac{∂L}{∂\dot{q}}=\frac{∂L}{∂q}$$

## The Attempt at a Solution

So since the masses are glued to the hoop, the radius is constant. So I am assuming my professor is referring to the oscillations in the angle? Meaning the hoop will just turn back and fourth?

So going by that (r=constant) My lagrangian is..
$$L=\frac{1}{2}mr^2 \dot{\theta}^2+2mr^2 \dot{\theta}^2+mgr(cos\theta +cos2\theta)$$
Which can be simplified obviously. I left it this way so it is easier to see my logic in the formulation of the Lagrangian? Is this correct? For the second mass, I used 2θ.

Now when I take the respective derivatives I get..
$$\ddot{\theta}-\frac{5g}{9r^2}\theta=0$$

The problem is, this is not an oscillator! If that was a plus sign I would be good to go. Could anyone help me out in figuring out this problem? Where was my logic flawed? :\??

Thanks!

In the way you are choosing to define your coordinate system, is g negative? If so, I think you're good to go. If not, consider your Lagrangian; there may be a plus sign where there ought to be a minus, especially before the potential energy term. It all comes down to how you chose to define your system, but as long as you were consistent with it, you should obtain oscillatory motion.

*Sigh* I got it, I made an algebra error. How embarrassing! :p!
Thanks! :D

## 1. What is the Lagrangian Problem and how is it related to two masses on a massless circle?

The Lagrangian Problem is a mathematical concept used in classical mechanics to solve for the motion of a system of particles. In the case of two masses on a massless circle, the Lagrangian Problem helps us determine the equations of motion for the masses and their positions on the circle.

## 2. How is the Lagrangian Problem different from other methods of solving for the motion of particles?

The Lagrangian Problem differs from other methods such as Newton's laws or Hamiltonian mechanics in that it takes into account the total energy of the system, instead of just the forces acting on the particles. This allows for a more comprehensive understanding of the motion and behavior of the system.

## 3. What are the key components of the Lagrangian Problem when applied to two masses on a massless circle?

The key components of the Lagrangian Problem in this scenario are the masses of the particles, the radius of the circle, and the angular velocities and accelerations of the masses. These variables are used to calculate the kinetic and potential energies of the system, which are then used to derive the equations of motion.

## 4. Can the Lagrangian Problem be applied to more complex systems with multiple masses and forces?

Yes, the Lagrangian Problem can be applied to more complex systems with multiple masses and forces. It is a versatile mathematical tool that can be used to solve for the motion of any system as long as the necessary variables and equations are known.

## 5. What are some real-life applications of the Lagrangian Problem?

The Lagrangian Problem has many real-life applications in fields such as physics, engineering, and astronomy. It is used to model the motion of celestial bodies, design mechanical systems, and study the behavior of particles in various environments. It is also fundamental in the development of quantum mechanics and other advanced theories in physics.