A weird answer -- Lagrangian mechanics

In summary: So you did what?You set ##\dot{\phi}=\text{const}## in the Lagrangian and then evaluated the EoM for ##\theta##.Yeah, yeah, I did that, and I got a result that makes sense. I was just saying...yeah.
  • #1
Andreas C
197
20
Just refer to my profile picture to see what the issue is! :biggrin:

Here's the problem: a ball of mass m is connected to a vertical pole with an inextensible, massless string of length r. The angle between the string and the pole is θ. The pole rotates around the z axis with a constant angular velocity $$\dotφ$$

We can easily determine the following equations:
$$x=r*sinθ*cosφ$$ $$y=r*sinθ*sinφ$$ $$z=r(1-cosθ)$$

Now I determined that the pesky Lagrangian is equal to the following:
$$L=\frac{mr^2}{2}(\dot\theta^2+\dot\phi^2sin^2\theta)+mgr(1-cos\theta)$$

So, where's the issue? Well, solving the Euler-Lagrange equations we find this:
$$mr^2\ddot\phi sin\theta+2mr^2\dot\phi\dot\theta cos\theta sin\theta=0$$

Which means that
$$\ddot\phi=-\frac{2\dot\phi\dot\theta}{tan\theta}$$

That's a really odd result. φ' is supposed to be constant, so φ''=0. But here we can see that φ'' can only be 0 if either φ' or θ' are 0, or if θ=π/4. Weird.

Am I missing something (I obviously am, it's a rhetorical question)?
 
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  • #2
You have to set
$$x=r \cos (\omega t) \sin \theta, \quad y=r \sin(\omega t) \sin \theta, \quad z=r (1-\cos \theta),$$
evaluate ##T=m \dot{\vec{x}}^2/2## and then derive the equation of motion for ##\theta## via the Euler-Lagrange equations.
 
  • #3
vanhees71 said:
You have to set ##\dot{\phi}=\omega=\text{const}## in the Lagrangian. Then you can evaluate the EoM for ##\theta##.

So I don't have to solve the Euler-Lagrange equations for φ and ##\dotφ## all?
 
  • #4
I've reformulated my suggestion a bit, but it's of course finally the same. If I understand your problem right, it's forced to have a rotation with constant angular velocity ##\omega=\dot{\phi}## around the ##z## axis. That's a constraint rather than a degree of freedom.
 
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Likes Andreas C
  • #5
vanhees71 said:
I've reformulated my suggestion a bit, but it's of course finally the same. If I understand your problem right, it's forced to have a rotation with constant angular velocity ##\omega=\dot{\phi}## around the ##z## axis. That's a constraint rather than a degree of freedom.

Ah ok! Thanks!
 
  • #6
Oh whoops, in my original post I meant "if θ was π/2", not π/4.
 
  • #7
As I said, you must enforce that ##\dot{\phi}=\text{const}##!
 
  • #8
vanhees71 said:
As I said, you must enforce that ##\dot{\phi}=\text{const}##!

Yeah, yeah, I did that, and I got a result that makes sense. I was just saying...
 

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of particles and systems in physics. It is based on the principle of least action, which states that the path a system takes between two points is the one that minimizes the action, or the integral of the Lagrangian function.

2. How is Lagrangian mechanics different from Newtonian mechanics?

While Newtonian mechanics focuses on forces and acceleration, Lagrangian mechanics takes a more abstract approach by looking at the energy of a system. It also does not require the concept of force, making it more versatile and applicable to a wider range of systems.

3. What is the significance of the Lagrangian in this framework?

The Lagrangian is a function that encapsulates all the information about a system's dynamics. It takes into account the potential and kinetic energy of a system, as well as any constraints that may affect its motion. From this function, the equations of motion for the system can be derived.

4. What are the advantages of using Lagrangian mechanics?

One advantage of Lagrangian mechanics is that it allows for a more elegant and concise approach to solving problems in physics. It also provides a deeper understanding of the underlying principles governing a system's motion. Additionally, it can be used to describe complex systems that may be difficult to analyze using traditional Newtonian mechanics.

5. Can Lagrangian mechanics be applied to all systems?

While Lagrangian mechanics is a powerful framework, it may not be suitable for all systems. It is most useful for systems with a small number of degrees of freedom, such as particles or rigid bodies. It may not be as effective for systems with a large number of interacting particles or for systems that involve quantum mechanics.

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