When can the Lagrangian be used

  • Thread starter dsdsuster
  • Start date
  • Tags
    Lagrangian
In summary, the conversation discusses the use of Lagrangian in cases where constraint forces do no work. The scenario of a bead threaded on a frictionless rod spinning with constant angular velocity is given as an example. The Lagrangian equations of motion are found to be the same as if the bead were subjected to a radial centripetal force. The question of why Lagrangian can be used in this case is raised, considering that the constraining force appears to do work. However, it is noted that centripetal forces, in general, do no work, even though the radius of the circular path may be changing.
  • #1
dsdsuster
30
0
Hello. I have a question about when the Lagrangian can be used. In the textbook we are using it is shown that for constraint forces which do no work, the Lagrangian is of the form T-U. In this question though, it seems like rod would provide a normal force that is in the same direction as the bead's instantaneous velocity at all times.

A bead of mass m is threaded on a frictionless, straight rod which lies in a
horizontal (x-y so no change in gravitational potential energy of the bead) plane and is forced to spin with constant angular velocity ω about a vertical axis through the midpoint of the rod.

Writing out the Lagrangian equations of motion, I get the same result as if the bead were always subject to a radial centripetal force mrw^2. I would basically like to know why the Lagrangian can be used in this case, when the constraining force seems to do work.

Thanks for your help!
 
Physics news on Phys.org
  • #2
Centripetal forces do no work.
 
  • #3
The radius of the circular path the bead is taking is changing though. Doesn't this imply work is being done?
 

1. When is the Lagrangian method used in physics?

The Lagrangian method is used in physics when solving problems involving motion of particles or systems, especially in classical mechanics. It is a mathematical framework that allows for a more efficient and elegant way to describe the motion of particles and systems.

2. What is the difference between Lagrangian and Hamiltonian mechanics?

Lagrangian mechanics and Hamiltonian mechanics are two different approaches to solving problems in classical mechanics. While the Lagrangian method uses generalized coordinates and velocities, the Hamiltonian method uses generalized coordinates and momenta. Both methods are equivalent, but the choice of which to use depends on the problem at hand.

3. Can the Lagrangian method be applied to systems with constraints?

Yes, the Lagrangian method can be applied to systems with constraints. In fact, this is one of its strengths, as it allows for the inclusion of constraints without the need for additional equations of motion. The constraints can be incorporated directly into the Lagrangian function.

4. Is the Lagrangian method only applicable in classical mechanics?

No, the Lagrangian method can also be used in other areas of physics, such as quantum mechanics and field theory. In these cases, the Lagrangian function may take on a different form, but the underlying principles and techniques remain the same.

5. What are the advantages of using the Lagrangian method?

The Lagrangian method offers several advantages over other methods in classical mechanics. It allows for a more concise and elegant representation of the equations of motion, and it is also more flexible in handling complex systems with constraints. Additionally, it is well-suited for problems involving multiple particles or systems, as it can easily be extended to include them.

Similar threads

I Forces of Constraint: Solving Euler-Lagrange Equation
    • Classical Physics
Replies
5
Views
1K
Constraint force using Lagrangian Multipliers
    • Advanced Physics Homework Help
Replies
2
Views
825
I Work done "against centrifugal force"
    • Classical Physics
Replies
9
Views
1K
I Using reference frames for derivation of Lagrangian
    • Mechanics
Replies
25
Views
1K
I Checking if a stationary point is a minimum using Lagrangian Mechanics
    • Mechanics
Replies
1
Views
897
Bead sliding on a uniformly rotating wire
    • Mechanics
2
Replies
37
Views
3K
I Making Eulers eqs. comply with Lagrange eqs.
    • Classical Physics
Replies
8
Views
1K
I Origin of Centripetal force when the net force toward the center is 0?
    • Classical Physics
Replies
7
Views
790
How do we provide centripetal force in this situation?
    • Mechanics
Replies
16
Views
1K
Spring-mass system with a pendulum using Lagrangian dynamics
    • Advanced Physics Homework Help
Replies
15
Views
2K

Hot Threads

Recent Insights

Back
Top