Lagrangian for two identical rods connected by frictionless joint.

In summary, the problem involves setting up a Lagrangian for a system with two rods and two angles as the degrees of freedom. The second rod's velocity is dependent on the motion of the first rod, making it difficult to find an expression for its kinetic energy. One attempt was to express the position of a mass element in cartesian coordinates and integrate to get the kinetic energy, resulting in an expression involving the parallel axis theorem for the moment of inertia. However, it is suggested to think in terms of the two angles as independent rotations instead.
  • #1
center o bass
560
2

Homework Statement


Hello. I have a problem with setting up the lagrangian for a system here.
The problem is stated at page 8 problem 2.3 with a diagram at the following

-->link<---


2. The attempt at a solution
I used two generalized coordinates corresponding to the angle between the first rod
and the roof as well as the smalles angle between the two rods.
My probem here lies in finding an expression for the kinetic energy.
The second rods velocity is dependent on the motion of first rod so I don't see any simple
moment of inertia relations to use here.

My first attempt was to express the possition of a little mass element dm on the second rod in terms of a vector in cartesian coordinates and then differentiate and square it to get that elements kinetic energy followed by integrating along the length of the rod to sum up the kinetic energy for each mass element. I then ended up with the following expression for the kinetic energy of the second rod:

[tex] K_2 = \frac{1}{2}m \left( \frac{1}{12}l^2 (\dot{\phi} + \dot{\theta})^2 + l^2 \dot{\theta}^2\right).[/tex]

I'm not too confident here. And even if the expression is right, I suspect there is an easier way to do it. Any suggestions?
 
Physics news on Phys.org
  • #2
You've identified the two angles as the two degrees of freedom. You should therefore not think in terms of movement of the rods, but rather in terms of these angles. Clearly the rotations about the angles are independent (the bottom rod should rotate at a constant speed, no matter what the top angle does).

Try using the parallel axis theorem to calculate the moment of inertia.
 

FAQ: Lagrangian for two identical rods connected by frictionless joint.

What is the Lagrangian for two identical rods connected by frictionless joint?

The Lagrangian for two identical rods connected by frictionless joint is a mathematical representation of the system's energy and motion, taking into account the kinetic and potential energies of the rods and their interaction with each other. It is typically denoted as L and is defined as L = T - V, where T is the total kinetic energy and V is the total potential energy of the system.

How is the Lagrangian derived for this system?

The Lagrangian for two identical rods connected by frictionless joint can be derived using the principle of least action, which states that the motion of a system will follow the path that minimizes the action (defined as the integral of the Lagrangian over time). By varying the Lagrangian with respect to the system's coordinates, the equations of motion can be obtained.

What are the constraints on the Lagrangian for this system?

The constraints on the Lagrangian for two identical rods connected by frictionless joint include the rod lengths, which must be constant, and the fact that the joint is frictionless, meaning there is no torque or force acting at the joint. These constraints are taken into consideration when deriving the equations of motion from the Lagrangian.

Can the Lagrangian be used to solve for the motion of this system?

Yes, the Lagrangian can be used to solve for the motion of the system by applying the Euler-Lagrange equations. These equations describe how the system will evolve over time based on the Lagrangian, constraints, and initial conditions. By solving these equations, the position, velocity, and acceleration of the rods can be determined at any given time.

What are the advantages of using the Lagrangian approach for this system?

The Lagrangian approach offers several advantages for analyzing the motion of a system, including being able to handle complex systems with multiple degrees of freedom and incorporating constraints and forces in a systematic way. It also provides a more elegant and intuitive way of understanding the system's motion compared to traditional methods, such as using Newton's laws of motion.

Back
Top