What do P, p(subscript r), and L represent in the 2-Body Lagrangian problem?

[dyn]

I have been looking at the problem of 2 point masses connected by a spring in polar coordinates. The problem is solved using the center of mass coordinate R and the relative coordinate r where M=total mass and m=reduced mass. The Euler-Lagrange equations then give equations for P(a vector) and p(subscript r) and L(which i think is p subscript θ). But i don't understand what these 3 quantities represent ?
Is it motion about the origin or the center of mass ? Is it in the radial direction ? p subscript r normally is but this time r is the relative coordinate , not necessarily radial.
Thanks

 

[voko]

The very first thing to do when using Lagrangian formalism is to determine your degrees of freedom.

Then introduce coordinates that express those degrees of freedom most naturally.

 

[dyn]

The center of mass coordinate has 2 degrees of freedom (R , θ ) and the relative coordinate has 2 degrees of freedom ( r and some other angle, i don't know what this is)

 

[voko]

If the system is confined to a plane, I agree. But is it?

Anyway, assuming it is planar, what do you not understand here?

 

[jhae2.718]

If the system is planar, then you should obtain four differential equations of motion from forming the system Lagrangian and applying the Euler-Lagrange operator.

 

[dyn]

yes the system is planar but i don't understand what the quantities found by solving the E-L equations represent. I know what is the momentum of the center of mass but what are the others ?

 

[jhae2.718]

You should get a system of four (probably nonlinear) ODEs which you can solve (most likely numerically) for your generalized coordinates as a function of time.

The Euler-Lagrange operator applied to the Lagrangian will give you an equation of motion. You can check your result by using another method, e.g. Newton, Hamilton, Kane, or Gibbs-Appell.

 

[voko]

yes the system is planar but i don't understand what the quantities found by solving the E-L equations represent. I know what is the momentum of the center of mass but what are the others ?

The quantities found from the E-L equations are the same quantities you used to compose the expressions for kinetic and potential energies for the Lagrangian function. If you do not understand what they are, then how did you obtain your Lagrangian?

 

[dyn]

I don't understand what the difference is between the generalised momentums of the centre of mass and the relative coordinate ? And what point is the angular momentum of the relative coordinate about ?

 

[voko]

These details depend on how exactly the coordinates were introduced. In one approach, the relative $r$ coordinate would be the distance from one mass to the centre of mass. And the relative $\phi$ coordinate would be the angle the mass-to-mass line makes with come fixed direction, say the vertical if the plane of motion is vertical, or the North-South line if the plane is horizontal.