Lagrangian of the system of two masses

[rouge89]

I am wondering, how does lagrangian of such system look like?

Will it be:

$$L=\frac{m_{1} \cdot \dot{y}^2}{2} + \frac{m_{2} \cdot \dot{x}^2}{2} +\frac{m_{3} \cdot (\dot{y'}^2+\dot{x'}^2)}{2} + \frac{I \cdot \dot{ \alpha }^2}{2} - mgy - mgy'$$

where:

$$y'=\frac{l}{2}sin(\alpha)$$
$$x'=\frac{l}{2}cos(\alpha)$$ ?

 

[andrien]

x^2+y^2=l^2,so x,y,ω(angular velocity)can be obtained.by the way that is correct

 

[Natey213]

I'm going to sit down and do this when I get a chance. But for now, it looks like you are only going to need one generalized coordinate to completely define the system. I would use the y-coord of m1.

 

[rouge89]

Hmm so guys, this is correct (with this additional equation as, andrien wrote: x^2+y^2=l^2), but there is way to do that better using one generalized coordinate, right? Hmm very often this one generalize is an angle + length... but here maybe it could be done with y,x etc.

 

[PJC01]

The Lagrangian of a system of two masses can be represented as the sum of the kinetic energy and potential energy of the system. The kinetic energy is given by the sum of the kinetic energies of each individual mass, which is equal to the mass multiplied by the square of its velocity. Therefore, the first two terms in the Lagrangian expression represent the kinetic energies of the masses m1 and m2.

The next two terms represent the kinetic energy of the third mass m3 and the rotational kinetic energy of the system, which is represented by the moment of inertia I and the square of the angular velocity α.

The last two terms represent the potential energy of the system, which is the sum of the gravitational potential energies of the masses m1 and m2.

The variables y' and x' represent the positions of the third mass m3, which is connected to the other two masses through a rod of length l and angle α. The potential energy terms take into account the vertical displacement of the masses due to the gravitational force.

Overall, the Lagrangian of this system takes into account the kinetic and potential energies of all three masses, as well as the rotational motion of the system. It is a comprehensive representation of the system's dynamics and can be used to derive the equations of motion for the masses.