Double Atwood Machine: relation between the contraints & the variables ?

[bobmerhebi]

Double Atwood Machine: relation between the contraints & the variables ?!

Hello. I am taking an analytical mechanics course & there's 2 "simple" equations relating the constraints to the variables. The problem is actually a class example. Here is it

1. The figure of the example is attached.

We are supposed to find the Lagrangian L = T-V, but I was stuck at correctly proving the following (for which I am asking for help in proving). I should note that I was able to find L & the equations of motion by altering the figure & making use of the height of the masses relative to (my) chosen reference.:

(x₂ + x₁) - l =0 && (2x₁ + x₂ + x₃) - (2l + l') =0

2. NO Relevant equations: simple arithmetic



3. The Attempt at a Solution :

Apparently l = x_p + x (though another element should be added -- can you tell me why this cannot be added ? -- & that is: \pi a, where a is the radius of the pulley).

Moreover,

x₃ = (l' - x') + x_p = (l + l') - (x₁ + x')

 

[bobmerhebi]

well i now know how this turns out to be. simple said, the radii of the pulleys are assumed to be negligible with respect to the length of the strings.