Is this correct? (Lagrangian mechanics)

[Lavabug]

Homework Statement

Find the Lagrangian and solve for the equations of motion. (see attached image)

Homework Equations

Euler-Lagrange's equation.

The Attempt at a Solution

I just wanted to check if I picked the right generalized coordinates. The pic shows what I think is called a Thompson-Tait pendulum. I chose the red dot as my reference frame and chose these as my generalized coordinates: the angles each of the pendulums at the ends make and phi as the angle at the center of pendulum.

I got a total of 3 holonomous constraints: z1, z2 = constant(all motion is planar), and (L + a)^2 = x^2 + y^2 for both pendulums at the ends.

Are my choices correct? Its a monster of a Lagrangian.

 

[mark726]

Yes, your choices for generalized coordinates are correct. The Thompson-Tait pendulum is a system with 3 degrees of freedom, so you have chosen 3 independent variables to describe the system. Your 3 holonomic constraints are also correct, as they represent the geometric constraints of the system (planar motion and the length of each pendulum).

As for the Lagrangian, it will indeed be quite complicated due to the presence of multiple pendulums and their associated kinetic and potential energies. But as long as you have correctly identified all the relevant variables and constraints, your Lagrangian should be correct.

To solve for the equations of motion, you can use Euler-Lagrange's equation as you mentioned. This will give you a set of coupled second-order differential equations, which can then be solved numerically or analytically (if possible).

Good luck with your calculations!