Conservation of ang. momentum using Euler-Lag. Eqns (help ;-)

In summary, The conversation discusses a proof about a trajectory and the use of Euler-Lagrange equations to show that the angle theta remains constant for all times t. The conversation includes a discussion about the potential energy involved in the problem and potential methods for solving the equation. The final conclusion is that the statement is not true for all potential energies but can be proven to be constant for a specific potential energy.
  • #1
T-7
64
0
Hi,

I'm stuck on what I suspect is a really easy proof. I'd appreciate some rapid help!

Homework Statement



For a trajectory with given initial position and velocity, we can always rotate the system of coordinates such that initially [tex]d\theta / dt = 0 [/tex] and [tex]\theta= \pi/2[/tex] say, t = 0.

Using the Euler-Lagrange equations, show that [tex]\theta[/tex] remains at [tex]\pi/2[/tex] for all times t.

The Attempt at a Solution



Having obtained the Lagrangian (L=mv^2/2 + U, rewriting v as [tex]\dot{r}^{2}+r^{2}(\dot{\phi}^{2}sin^{2}\theta + \dot{\theta}^{2})[/tex] in spherical coordinates, and taken partial derivatives with respect to [tex] \dot{\theta} [/tex] and [tex] \theta [/tex], and [tex]\phi[/tex] and [tex]\dot{\phi}[/tex], I have found the following relations (I think they're correct?)

[tex]d/dt [mr^2\dot{\theta}] = mr^2\dot{\phi^2}cos(\theta)sin(\theta)[/tex]

[tex]d/dt [mr^{2} \dot{\phi} sin^{2} (\theta)] = 0[/tex]

(There is a third expression, from the partial derivatives wrt. [tex]r[/tex] and [tex]\dot{r}[/tex], but I doubt it's relevant here.)

I'd be tempted to integrate both sides of the first to get an expression for [tex]\dot{\theta}[/tex], but that's where I become unstuck just now... I'm not sure how to integrate the RHS with respect to time, given what I know about the components. But presumably integration is going to be involved here, or the initial conditions would never come into play (?). I can deduce from the second that [tex]mr^{2} \dot{\phi} sin^{2} (\theta) = const = c[/tex], and so at t = 0 [tex]mr^{2} \dot{\phi} = c[/tex], but nothing more of interest seems to be following at the moment...

Incidentally, I take the geometrical meaning to be that the motion is confined to a plane orthogonal to L, ang. momentum is conserved.

Advice gratefully received (as soon as possible!) Many thanks :-)
 
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  • #2
The statement you are trying to prove is not true for any potencial energy U. Does the problem include something like U=U(r)?
 
  • #3
Lojzek said:
The statement you are trying to prove is not true for any potencial energy U. Does the problem include something like U=U(r)?

Yep. :-)
 
  • #4
T-7 said:
[tex]d/dt [mr^2\dot{\theta}] = mr^2\dot{\phi^2}cos(\theta)sin(\theta)[/tex]

If you put theta(t)=pi/2, then the equation is always true. I think this proves that theta must be constant, since a physical system can not have more than one solution.
 

Related to Conservation of ang. momentum using Euler-Lag. Eqns (help ;-)

1. What is the conservation of angular momentum?

The conservation of angular momentum is a fundamental law of physics that states that the total angular momentum of a system remains constant unless acted upon by an external torque. This means that the total amount of rotational motion in a system will remain the same unless an outside force is applied.

2. What are the Euler-Lagrange equations?

The Euler-Lagrange equations are a set of equations used to describe the motion of a system in terms of its coordinates and velocities. They are derived from the principle of least action, which states that the path taken by a system between two points will be the one that minimizes the action of the system.

3. How are the Euler-Lagrange equations used to conserve angular momentum?

The Euler-Lagrange equations can be used to describe the conservation of angular momentum by incorporating the concept of torque into the equations. This allows us to see how changes in the system's coordinates and velocities will affect the amount of angular momentum in the system.

4. Can the Euler-Lagrange equations be applied to all systems?

Yes, the Euler-Lagrange equations can be applied to any system that can be described in terms of coordinates and velocities. This includes both classical and quantum mechanical systems, making the equations a powerful tool in understanding the conservation of angular momentum in a wide range of scenarios.

5. What are some real-world applications of conservation of angular momentum using Euler-Lagrange equations?

The conservation of angular momentum is a crucial concept in many areas of physics and engineering. For example, it is used in understanding the motion of planets and satellites in space, the behavior of spinning objects such as gyroscopes, and the design of vehicles like bicycles and helicopters. The Euler-Lagrange equations provide a mathematical framework for analyzing these systems and predicting their behavior.

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